Le MSU BETURNING MATERIALS: Place in book drop to [JBRARJES remove this checkout from n. your record. FINES win be charged if book is returned after the date stamped be10w. THE GEOMETRY OF MULTIDIMENSIONAL SCALING BY Daryl W. Tingley A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1980 ABSTRACT THE GEOMETRY OF MULTIDIMENSIONAL SCALING BY Daryl W. Tingley Multidimensional Scaling (M.D.S.) is a process of finding representations of a given class of objects as points in a metric space. The metric represents the similarity (or dissimilarity) between Objects of the given class. Examples of M.D.S. are the representations of colors as points in a metric space, and of pure tones as points on a helix. Once such a representation has been found, it is natural to asx if the representation is, in some sense, unique. The bulx of this thesis is devoted to the study of uniqueness questions. An order transformation f between two metric spaces M and M is a function such that l 2 That is, order transformations preserve "the order of the distances". The uniqueness question can be stated as follows: If f :Ml-aM2 is an order transformation, is f necessarily a similarity? In general, the answer is no, and examples are given in Chapter 1. However, by considering only certain types of spaces, such as subsets of En, we demonstrate many situations where f is necessarily a similarity. Many of the results in this thesis are valid for the more general metric transformation.‘ A function between two metric spaces is called a metric transformation if it preserves equality of distances. Typical theorems in this thesis are: Theorem: If M1 and M are convex metric spaces, and 2 f is a bicontinuous metric transformation from M1 onto M2, then f is a similarity. Theorem: Let Ss:Em, m 2_2, be a connected set with non-empty interior. Then any metric transformation of S into En is either a similarity, or maps S onto a single point. In Chapter 6 some results are Obtained about the existence of order transformations. Typically these results discuss whether or not a given metric space can be order embedded into another metric space. ACKNOWLEDGMENTS This author wishes to express his gratitude to Professor L. M. Kelly for his helpful suggestions and guidance during the research. In addition, he would lixe to thanx the guidance committee, Professor Dubes, Professor Ludden, Professor Moran, and Professor Sonneborn, for their many helpful suggestions and comments. ii CHAPTER 1. CHAPTER 2. CHAPTER 3. TABLE OF CONTENTS LIST OF FIGURES . . . . LIST OF SYMBOLS . . . LIST OF DEFINITIONS . . INTRODUCTION . . . . . PRELIMINARIES AND EXAMPLES §l. Definitions . . . . . . . §2. Definitions of Order and Metric Transformations §3. Formal Statement of the Problem §4. Examples . . . ... . . . RESULTS IN GENERAL METRIC SPACES §1. Elementary Lemmas §2. On the Continuity of Metric Transformations §3. Metric Transformations Between Convex Sets §4. A Relationship Between the Curvatures of an Arc and of its Image Under a Metric Transformation RESULTS IN NORMED LINEAR SPACES §l. Definitions and Elementary Lemmas §2. On the Continuity of Metric Transformations Between Subsets of Normed Linear Spaces 93. Metric Determinacy 2h) Normed Linear Spaces iii PAGE vi 55 57 66 72 CHAPTER 4. CHAPTER 5. CHAPTER 6. n ORDER AND METRIC TRANSFORMATIONS IN E §1. §2. §3. §4. §5. A Discussion of Metric Bases On Metric Transformations Between Subsets of Euclidean Spaces n Screw Lines in E ... . . Order Transformations of the Unit Sphere in En Appendix to Chapter 4 FURTHER RESULTS IN EUCLIDEAN SPACE g1. §2. Notation, Definitions, and . . . . Elementary Lemmas Metric Transformations Between Hypersurfaces in En EXISTENCE QUESTIONS . §l. On the Existence of Order . Embeddings of Finite Sets Into En §2. On the Existence of Order Embeddings of Spheres Into Normed Linear Spaces CONCLUSION BIBLIOGRAPHY iv PAGE 84 86 89 101 122 131 133 136 143 152 152 162 171 LIST OF FIGURES FIGURE PAGE 19 21 N r4 rd H a: rd nu w 34 64 .1 . . . . . . . . . . . . . . . . . . . . . . . . 91 125 127 128 bbhbbbb \lO‘U'I-P- . 129 . 140 U1 ..J .1 . . . . . . . . . . . . . . . . . . . . . . . . 156 .2 . . . . . . . . . . . . . . . . . . . . . . . . 157 . 159 .4 . . . . . . . . . . . . . . . . . . . . . . . . 163 .5 . . . . . . . . . . . . . . . . . . . . . . . . 167 O‘C‘O‘O‘O‘O‘ w .6 . . . . . . . . . . . . . . . . . . . . . . . . 168 aff A s s B(x,r) LIST OF SYMBOLS See page 165 Real numbers. See page 104 The open ball with center x and radius r A class of sets The complex numbers n-dimensional complex space Theoretic set difference. See page 28 The closure of the set C The distance set of N. See page 14 The distance between x and y The differential of N at the point p, called the Weingarten Map The Euclidean plane n-dimensional Euclidean space Metric transformations The function f restricted to the set C Hilbert space. See page 11 n-dimensional hyperbolic space The identity transformation vi KH(y.p) . . . . . . . The Haantjes-Finsler curvature of the arc y at the point p. See page 49 K (p) . . . . . . . The classical curvature of the Y arc y at the point p. See pages 49, 137 xN(p,v) . . . . . . The normal curvature of a given hypersurface at the point p and in the direction v £(y) . . . . . . . . The length of the arc y 1(P) . . . . . . . . See page 31 L (q,r) . . . . . . . The length of the arc y from Y the point q to the point r I; See page 12 £2 See page 12 (M,d),(M1,d1),... Metric spaces M.D.S. . . . . . . . Multidimensional Scaling mesh(P) . . . . . . . See page 43 N.L.S. . . . . . . . Normed linear space (or spaces) (N,d),(N1,d1),... Distance spaces N(p) . . . . . . . . The unit normal to the given hypersurface, at the point p. and in the given orientation n(p,a) . . . . . . . The unit normal to the arc a at the point p p,p1.p’,q,... Points in a metric or distance space pq . . . . . . . . . The line through the points p and q (p,q) . . . . . . . . The open segment from the point p to the point q [p,q] . . . . . . . . The closed segment from the point p to the point q vii H'O (DU) '0 S(p.r) SxS US,U§,... U1,U2,...U7 V'V-pooo V‘ V'Vi'vij"°' HVH (v1.v2). v ] [v1,..., m W,Wj,... A scale function. See page 14 The real line n—dimensional spherical space The space tangent to the hypersurface S at the point p The sphere about the point p with radius r The cartesian product of the set S with itself Real numbers. See page 104 A set or a transformation Affine transformations Usually angles A set or a transformation Unitary transformations See page 105 Vector spaces The subspace perpendicular to the subspace V Usually vectors. See page 104 or 138 The norm of the vector v The inner product of the vectors v and v 1 2 The space spanned by the vectors v ... v 1' ' m Vector spaces Usually a vector. See page 104 Usually vectors viii Usually vectors. See page 104 Usually vectors. See page 104 See page 137 See page 137 An equation. See page 85 or 89 1X LIST OF DEFINITIONS Affine, similarity, S7 subspace, 57 transformation, 57 Arc,31 geodesic, 32 length of,31 rectifiable, 31 Ball, unit, 59 Basis, metric, 86 Bicontinuous, lO C-metrized space, 3 Continuous, 10 Converge, 10 Convex, 4O algebraically, 58 pseudo, 41 strictly algebraically, 58 Curvature, classical, 49 Haantjes-Finsler, 49 normal, 137 Discrete space, 24 Distance, set, 14 space, 9 Equilong map, 138 Extreme point, 165 Face, 165 Facet, 165 Flat, m-flat, ll Fundamental form, 137 Geodesic arc, 32 Isometric, 10 order, 12 Isometry, lO Isomorphism, order 13 Join, 31 Line, 31 in a N.L.S., 58 Long legged local isosceles property, 34 m-flat, 11 Mesh, 43 Motion, 87 Metric, basis, 86 space, 9 transform, 13 transformation, 13 transformation, trivial, 13 Normally ordered set, 31 One parameter subgroup of motions, 108 Open function, 73 Order embeddable, 12 embedded, 13 isometric, 12 isomorphism, 13 transformation, 12 Rectifiable arc, 31 Relative interior point, 166 Scale function, 14 Scalene configuration, 155 Screw curve or line, 20 Segment, 31 algebraic, 58 Similar, 10 Similarity, 10 affine, 57 Space,C-metrized, 3 distance, 9 metric, 9 Span, 86 Spread, 14 Stress, 14 Support, 165 Supporting set, 165 Transform, metric, 13 Transformation, affine, 57 metric, 13 order, 12 unitary, 105 Triangle equality, 62 Trivial metric transformation, 13 Ultra metric inequality, 154 Unit ball, 59 Unitary transformation, 105 Weingarten map, 137 Xi INTRODUCTION The various psycho-physical theories associated with color, sound and other sense perceptions have given rise to many efforts to "represent" perceptual dissimilarities as distances in a conventional geometric structure. Newton [30], Helmholtz [l6], Schrodinger [33], and Henning [17], are but a few of the scientific figures who have contri- buted to the geometrization of psycho-physical perception theories. More recently a school of perception theorists led by the contemporary psychologists Torgerson and Shephard have advanced more purely psychological theories explicitly distinguishing between the physical properties of light and the subjective sensations which it produces. These theories have spawned a technique known as multidimensional scaling with applications far beyond the boundaries of perception theory. It is with this notion of multidimensional scaling (MDS) that this thesis is concerned. Some sense of the explicitly geometrical thinking of the Shephard-Torgerson school can be gathered from the following introductory remarks taken from a recent paper by J.P. Cunningham and R.N. Shephard ['7]entitled "Monotone Mappings of Similarities onto a General Metric Space". STATEMENT OF THE PROBLEM "In ordinary, nontechnical discussions of the perceived similarities and differences among things (whether faces, voices, tastes, odors, colors, etc.), we readily make use of a spatial metaphor. Thus we may say that one shade of color is very glggg pg or, alternatively, §§£_;;gm another; or even that one shade seems to be somewhere between two others. This automatic use of clearly spatial terms such as "near", "far", or "between" to characterize subjective similarities and differences suggests that there is an implicit connection, in human cognition, between the con- cepts of similarity and dissimilarity, on the one hand, and the concepts of spatial proximity and distance on the other. Specifically, it suggests that similarities among objects are related by some sort of monotone decreasing function to distances among points corresponding to those objects in some sort of metric space." As a first step in trying to determine the nature of the presumed real number distance function of an individual's color perception space, much experimenting has taken the form of studying the responses when the individual is asked to decide which of two pairs of presented colors "is the better match". If all such comparisons are made for a given finite set of colors presumably some evidence is adduced about the presumed underlying metric. In the early days of such activity an effort was made to "realize" the resulting structure "on the blackboard" in such a way that the "black board distances" (presumably Euclidean 2-dimensiona1) reflected the order properties of the observed comparisons. This leads in the direction of the following abstrac- tions. If S is a set and C a chain (linearly ordered set) with minimal element 0, and if e is a mapping from S X S into C satisfying e(x,y) = e(y,x), e(x,y) = 0 if and only if x = y, then (S,C,e) is a C-metrized space. If C is a subset of the nonnegative real numbers, (S,C,e) is called a semi—metric, or distance space. The blackboard embedding problem is an attempt to "order embed" a given C-metrized set S into E2, that is to find a map 2 f: S a E such that e(x,y) g e(u,v) if and only if d(f(x)of(Y)) S d(f(u)of(v))- There is, or course, no reason to believe that such embeddings are always possible. In fact, if S contained four p01nts x1,x2,x3,x4 such that e(xi,xj) = e(xk,x1) for all permutations i,j,k,1 of l,2,3,4 then S could not be order embedded in E2. While the space (S,C,e) may 2 not order embed in E it may be possible to "nearly" embed it. That is, it may be possible to find a mapping of S into E2 such that there are relatively few rever- sals of the order of distances. This idea is at the root of a scheme divised by J.B. Kruskal of Bell Telephone Labs to produce the "best" approximate order embedding possible into a Euclidean space of specified dimension [20]. The measure of approximation is called the stress of the embedding and essentially measures the departure from strict monotonicity of the realized distances. Kruskal and others have concocted computer programs which make this embedding process quite simple and efficient. This thesis does not consider approximate order embeddings. There are a number of natural questions about order embeddings, the first of course being, is such an embedding possible? Then, is it unique, in some sense? What about the possibility of embedding in spaces other than B“? In practice the psychologist is trying to adduce the nature of the "underlying" metric by examining a large number of finite subsets. Is there any reason to suppose that, even if every finite subset of an "unknown" metric space is order embeddable in, say, E2, then the space itself must be order embeddable in E2? Broadly speaking, three questions suggest themselves. These questions will also be asked about metric embeddings (see Chapter 1). I. Given two distance spaces N1 and N is N 2' 1 order embeddable into N2? We call this the existence question. II. Is the embedding unique, in some sense? We call this the uniqueness question. III. If each finite subset of N is order embeddable 1 into N2, what is the relation of N1 to N2? Question I would seem to be the most important. Note that there is no loss of generality in beginning with a distance space, rather than some C-metrized space (S,C,e). For if there is an order embedding f of (S,C,e) into a distance space (N2,d) there is induced by this embedding a one-to-one map from C to If? given by e(x,y) + d(f(x),f(y)). Thus, for there to be any hope of (S,C,e) being order embedded into N2, there must be a one-to-one mapping of C into Iii, and we may assume the minimal element of C goes to 0. Using this map, we may take C to be 12+. Nbrmally, an M.D.S. theorist would want N2 to be a familiar space, or a subset of a familiar space, such as En or a normed linear space. We will present some answers to questions of type I, although many of them are negative in that they show, for certain N1 and N there is no 2. order embedding of N1 into N2. Question II is very important in the applications of MDS to prdblems of data analysis. It is widely assumed that two different order embeddings of a distance space in En are "nearly" equivalent "up to a scale factor". This presumably means that they are "approximately" similar. But a moment's reflection shows that this is not literally true. Imagine a set in E2 with large (finite) cardinality and one additional point, very far removed from this set. Certainly this latter point can be rather freely perturbed without affecting the order of the distances, but the resulting set will not be even approximately similar to the original set. MDS theorists would characterize this set as excep- tional and maintain their continued faith in the "principle of metric determinancy" which asserts that in a "highly structured" space such as En the order of distances of a configuration "essentially" determines the configuration, if the cardinality of the configuration is "large" and the configuration is not "exceptional". There is strong intuitive reason to feel that there is an element of validity to this “principle" but its precise formulation let alone its proof is very elusive. One of our main purposes in this thesis is to examine the proposition that the order of distances of a configuration in a highly structured space determines the configuration. Unfortunately we can make no contribution to the third question. Results of this type would be extremely useful, for in practice the MDS user deals with finite subsets of infinite sets - such as a finite set of colors, and tries to determine the underlying distance function from this. This thesis is divided into 6 Chapters. To help orient the reader, and to indicate what we are trying to achieve we briefly outline these Chapters. Chapter 1, called Preliminaries, is used to introduce definitions and to make precise some of the ideas discussed in the Introduction. In particular, §2 formally defines the concepts of order transformation and metric transformation, while §3 further discusses the problems to be considered in this thesis. Chapter 1 concludes with a number of examples, designed to help the reader become familiar with order and metric transformations, and with the types of problems we consider. Chapter 2 investigates order and metric transformations in very general metric spaces. The results obtained form the foundations for much of the work in later chapters. Sections 1 and 2 primarily investigate the continuity of order and metric transformations. Section 3 investigates metric transformations between convex spaces. Much of the work on convex spaces is a reorganization of previously known results. Finally, §4 investigates the effect of a metric transformation on the curvature of a curve. This is done using a metric definition of curvature. Chapter 3 investigates metric transformations between subsets of normed linear spaces (N.L.S.). Section 1 consists largely of preliminaries such as definitions and known results. In the second section, we apply results from Chapter 2 and investigate the continuity of metric trans- formations between subsets of N.L.S. The main results of the chapter are presented in 93, and the following corollary is typical. Corollary_3.19: Let M1 be a N.L.S. of finite dimension at least 2, with a strictly convex unit ball. If Us:M1 is a set with a non-empty interior, then any metric transformation from U into a N.L.S. with the same dimension as M1, is a similarity. Chapters four and five investigate metric transformations between subsets of Euclidean spaces. Section 1 of Chapter four contains bacxground material. In section 2, a theorem of Schoenberg is extended. Schoenberg showed that any metric (or order) transformation of Em, m 2.2, into En is necessarily a similarity. We extend this to subsets of Em. In section 3, all metric (and order) transformations of the real line into En are characterized. This is an elaboration of a known result (von-Neumann and Schoenberg). Finally, section four of Chapter four investigates metric and order transformations from the unit sphere of En into arbitrary N.L.S. Chapter five considers metric transformations between hypersurfaces in En. Techniques of differential geometry are used, forcing us to assume that the metric transformation is differentiable. Finally, Chapter 6 briefly discusses "existence questions". That is, when do order transformations exist between two sets? There are two sections, the first of which deals primarily with finite sets, the second with infinite sets. CHAPTER 1 PRELIMINARIES AND EXAMPLES As indicated in the introduction, the principle concern of this thesis is with the problems of order and metric embeddings of one distance space into another, with emphasis on determining when such embeddings are unique. we nOW‘WiSh to make these concepts precise. §1. By a distance spagg is meant what is commonly called a semi-metric space, that is a set" N together with a mapping d: N x N 4 If? (non-negative) such that for (x,y) 6 N X N, d(x,y) = d(y,x) and d(x,y) = 0 if and only if x = y. For the most part, although not always, our distance spaces will also satisfy the triangle inequality, in which case the space is called a metric spagg, We denote a distance space by listing the set and the mapping symbol thus: (N,d), or simply by writing the set symbol N. Symbols used for distance spaces in this thesis are (N,d), (N’,d’), (Nl,d1), and (N2,d2). If the space is to also satisfy the triangle unequality the symbols (Mod): (M',d’), (M1,dl), (M2,d2) are used, and we also state explicitly that the space is to be metric. 10 As a metric space is a topological space, the concept of continuity is very familiar. We briefly discuss continuity in general distance spaces. A sequence {pi} in a distance space N is said to converge to p if and only if lim d(p,pi) = O. This is denoted by lim pi = p. A function f from N1 into N2 is said to be continuous a; .2 if and only if for any sequence {pi} with lim pi = p it is true that f(p) = lim f(pi). Considering f as a function 9329 its range, f is said to be bicontinuous if and only if f is inver- tible and both f and f-1 are continuous. The distance function d of a distance space N is said to be con- tinuous if and only if for any two sequences {pi} and [qi} with lim pi = p and lim qi = q it is true that lim d(pi,qi) = d(p,q). It is easily seen that a metric is continuous. Distance spaces with continuous distance functions are important for, unlike distance spaces in general, the sets of the form [xld(p,x) < 6, 6 > 0} form a basis for a topology on the set N. We refer the reader to Blumenthal [‘4] for more details. Two distance spaces, N1 and N2, are said to be isometric if and only if there is a function g from N1 onto N2 such that for x,y 6 N1, d1(x,y) = d2(g(x),g(y)). The function g is called an isometry. If g is such that for x,yENl, d1(x,y) =kd2(g(x),g(y)), k > 0 then N1 and N2 are said to be similar and g is called a similarity. 11 Two isometric distance spaces are, from a geometer's view- point, "the same", while two similar distance spaces are "the same except for the unit of measurement". We assume a familiarity with the elementary topology of metric spaces, including commonly used terms. Strictly geometric concepts such as arclength, geodesic, curvature, etc. are defined when they are first used. For the most part the terminology of Blumenthal [4.] is used. In addition to some topology, the reader should be familiar with the concept of a normed linear space, and a linear transformation for Chapter 3, some linear algebra, including some knowledge of the standard theorems on simultaneous diagonalization of linear transformations for Chapter 4, and a knowledge of the differential geometry of hypersurfaces in En for Chapter 5. We often look at very specific metric spaces. These include En - n-dimensional Euclidean space Sn - n-dimensional spherical space Hn - n-dimensional hyperbolic space. n . . By an m-flat of En,Sn, or H we mean an isometric image of Em,Sm, or Hm in En,Sn, or Hn respectively. N - Hilbert space. By this we mean all sequences Q x1, . . . ,x , . . . of real numbers such that Z x? n i=1 1 with the usual inner product. 12 We also consider normed linear spaces (N.L.S.). In particular, the spaces I; which consist of all n-tuples oereal numbers with the norm given by “(xl,...,xn)”p = (23linp)l/p and 1: whose norm is given by ”(Xl,...,xn)H¢ = mixflxil}, will be used. We refer the reader to Blumenthal [‘4] for a more detailed discussion of these spaces. §2. We now make precise the notion of order preserving transformation. Definition: If N1 and N are distance spaces and 2 f is a mapping from N1 into N2 such that for all x,y,u,v 6 N1 (1) d1(xoY) S d1 (111V) :9 d2 (f (X)of(Y)) S d2 (f (u) of(V)) then f is called an order preserving transformation or simply order transformation of N1 into N2. The image f(Nl) is said to be order isometric to N1 and N1 is said to be order embeddable into N 2. Some properties of order transformations follow easily from the definition. It is easy to see that if (1) holds for all x,y,u,v E N then 1 (2) dl(x.y) = dl 0 respectively. Note that isometries and similarities are order preserving. §3. In the introduction we said the purpose of this thesis is primarily to work on the "uniqueness" question, with some work on the "existence" question. We now try to be more explicit as to the meaning of these terms. The "existence" question asks: When is one distance space order embeddable into another? This can only mean the following: Given two distance spaces N1 and N2,‘ is there an order transformation f of N1 into N2? The scope of the question may be widened by asking: Given a distance space N1 and a class of distance spaces 0. is there an order transformation f of N1 pppp some member N2 of 6? Note the assumption of onto here, rather than into. If desired, 6' could be such that any subspace of any member of 6' is also in Cu hence the above question includes the same question with into, rather than onto,used. On the other hand, it may not be desirable for f(Nl) to be certain subspaces of N Thus phrasing the question as 2. we do allows more flexibility. The same types of questions can also be asked of metric transformations. The uniqueness question is harder to state, for it is not clear what "unique" means. If f and g are order transformations of N1 onto members of a class <3 of distance spaces we could ask if necessarily f 5 g. 16 However, this is too restrictive for our purpose. For example if C is all subspaces of En, f: N 4 f(N) g E is an order embedding, and h is a similarity of En onto itself, then hof is also an order embedding of N into En. As f(N) and h(f(N)) are similar, so essentially "the same" from a geometric view point, it would be more desirable to ask if f were unique, "up to a similarity". This could be phrased as follows: If f and g are order transformations of N onto members of a class of distance spaces Cu is there a similarity h such that h°f E 9? Order isometries are invertible, so in the above question we have h E gof_1. N w because 9 and f are order isomorphisms, gaf-l is also an order isomorphism and gof-l: f(Nl) 4 g(Nl). Thus equivalent to the above question is the following: If f is an order transforma- tion from N1 onto N N1 and N2 members of a class 2' O of distance spaces, is f a similarity? This phrasing is nice as we stay within the class 6' of spaces, yet the seemingly more general question (above) is answered. Either of the above two questions could be asked of metric transformations as well, however they would no longer be equivalent. (A metric transformation need not be invertible.) Most of our work will consider the second question, with metric rather than order transformations used. That is, we ask: If f is a metric transformation from N1 onto N2, N1 and N2 members of a class C. of 17 distance spaces, is f a similarity? Occasionally we will insist that f have further properties such as continuity, or differentiability, but We try to stay away from this, putting restrictions on the class 0, rather than on the function f. §4. At this point we wish to present some examples of order and metric transformations. Further examples are presented in appropriate Chapters - usually as counter- examples. The purpose of the following examples is to show that, even in "highly structured" spaces, the order of the distances of a set does not determine the set, even up to a similarity. In addition, in none of these examples is there any hope of formulating a notion of "approximate" similarity. Example 1 shows that, given any metric or distance space, there are many metric or distance spaces order isomorphic to it. Example 1 (Wilson) l39_[: Let p be any real valued function with domain the non-negative real numbers, and with the following properties 1) 9(0) = 0 2) p is strictly increasing 3) P(lt1+ (l-l)t2) 2 lp(tl) + (l-l)p(t2) for t1,t220,ogxg1. It is shown in [39] that if (M,d) is any metric space, 18 then (M,p°d) is also a metric space. That (M,d) is order isomorphic to (M,pod) is easy to see. The scale function of the order transformation (the function which assigns each point p of M to itself) is p. Example 3 is of this form. Condition (3) is used only in proving (M,pod) satisfies the triangle inequality. Thus if one were interested only in distance spaces, Condition (3) could be omitted. Example 2: Let S be a set of points on the line segment [qr], on the y-axis, and let p and p' be any two points on the x-axis, further than d(q,r) from the origin (see Figure 1.1) . Then S U {p} is order isomorphic to S U {P'}. The function f(x) = p’ x=p establishes the order isomorphism. See Figure 1.1. Example 3: A semicircle of radius r and a line segment of length 2 are order isomorphic (see Figure 1.2). The order isomorphism is f(x) = (r,e) where 9 = E?', and x and (r,9) are as in Figure 1.2. Example 4: The mapping from It to a helix given by t 4 (a cos t, a sin t, bt) where a and b are constants, is easily seen to be a metric transformation, whose scale 2 2 2 function is given by p2 (d) = 4a sin2(%) + b d . If b2.2 2a2, by checking the derivative of p, it is seen that f is an order transformation. 19 q«) p p’ r 0 Figure 1.1 (no) e\ +——-r -——§ 1 .¢ Figure 1.2 20 Example 5: Let C be the unit circle [(cos 6, sin e)lo g 6 < 2?]. We map C into E4 by f((cos 9, sin 9)) = (cos a, sin 9, %cos 29, %sin 29) If p = (cos 91, Sln 61) and q = (cos 62, Sln 92) then e -e d(p.q) =/4 sin2 (32—1) e -e d(f(p),f(q)) =/4 sin2 FAT—1') + E":"m2(°2"91) Because both d(p,q) and d(f(p),f(q)) depend only on the quantity I62-Gll, f defines a metric transformation of C into E4. Furthermore, by investigating the functions g(t) =./4 sinzt/Z and h(t) =./4 sinzt/2+sin2t , it can be shown that f is an order transformation. 4 of The helix, the circle, and the curve in E Example 5 are known as screw curves. A screw curve in a metric space M is a metric transformation of fit into M. von-Neumann and Schoenberg [29] characterized all continuous . n screw curves in E as those curves of the form (Alcos klt, A sin klt, A cos k t, A sin k t,...,ct) 1 2 2 2 2 where Ai'ki' and <: are all constants. We shall present a proof and extension of this in Chapter44. Example 6: Consider the space (lR,d) where ]R is the set of real numbers and d(x,y) =\/Ix-y|. It is easily seen that CR,d) is a metric space and is order isomorphic 21 to fit. It is shown in Blumenthal ([4-], Sec. 54) that this space is ppp isometric to any subset of En, for any n, but it is isometrically embeddable in Hilbert space. Note that the scale function of the order isomorphism, p(d) =\/d , has no spread, for lim‘é? does not exist. This space, daO considered as a subset of Hilbert space, is also interesting because it is a curve which has a tangent at no point. Example 7. Let M be either hyperbolic or spherical n-space. Let S c M be an m-flat in M, m,< n, and let a (a less than g x radius,for spherical space) be given. For 3 es, let f(s) be that point of M which is a distance a above S. (i.e. d(s,f(s) = o and the segment joining s to f(s) is perpendicular to S). Then f is a metric transformation for, if d(p,q) =d(p",q’) , it is easy to show the quadrilaterals qu(q)f(p) and p’q’f(q’f(p’) are congruent. f(p) f(r) f(q) 1 2 3 4 1 1 p r q Figure 1.3 22 To see that f is ppp a similarity, consider three distinct, collinear points p,q, and r in S, with d(p,r)4-d(r,q) = d(p,q). If f is a similarity, then d(f(p).f(r)) +d(f(r).f(q)) = d(f(P).f(q)) - Angles 1 and 2 (in the above diagram) are equal, as well as angles 3 and 4. (Using congruent triangles.) Thus a: 2+): 3={ 1+{ 4. In both spherical and hyperbolic space, the sum of the angles of a quadrilateral is ppp 360°. Hence { l-+{'2-+{’3-+{ 4 # 360°. So { 2-+{ 3 # 180°. Thus f(r) does not lie on the line through f(p) and f(q) and d(f(p),f(rH+d(f(r).f(q)) #d(f(p)+f(q)) . C] These examples show that in general the order of the distances of a distance space doesfl determine (even up to a similarity) the space. Even rather nice subspaces of En, such as Examples 3, 4, and 5 are not determined by the order of their distances. In none of these examples could one hope to say the two Spaces are "approximately" similar. One of our goals is to show that, at least in common well structured spaces, there may be some truth in the statement that these examples are exceptional. On the other hand, we hope any user of M.D.S. would recognize the limi- tations of MJD.S. shown by these examples. CHAPTER 2 RESULTS IN GENERAL METRIC SPACES In this Chapter results are obtained about order and metric transformations in general metric spaces. The results obtained are nice applications of metric geometry. The Chapter is divided into four sections. The first two deal, for the most part, with the continuity and bicontinuity of metric and order transformations. The first section contains a series of lemmas, all of which are easy to prove. The major result shows that any order transfor- mation between two distance spaces is necessarily bi- continuous, unless one of the two spaces is discrete. Some properties of the scale function are also developed. The second section considers the continuity of metric transformations. It gives sufficient conditions for a metric transformation to be continuous. The conditions are mostly on the domain and range of the transformations but, unlike the same problem for order transformations, some (quite rnild) conditions are imposed on f. The third section shows that in the class of convex metric spaces, the order of the distances determines the Space, up to a similarity. There is also a metric trans- formmation version of this and several results which relate 23 24 the arc length ci’ a curve, and the image of that curve by a metric or order transformation. Finally, the fourth section includes an interesting result relating the curvature of a curve and the curva- ture of the image of that curve by a metric, or order, transformation. §l. We begin Section 1 with a discussion of what is to be done. Let f be a metric or order transformation, from a distance space (N1,d1) to a distance space (N2,d2), with scale function p. Lemmas 2.1 and 2.2 show that if f is an order transformation, and neither Nl nor N2 is discrete, then f is bicontinuous. This is done by showing that necessarily 1im p(d) = 0, from which deo continuity follows easily. (A distance space N is said to be discrete if for each p E N, there is an e > 0 such that [x|d(p.X) < s} = [Pi-1 Lemma 2.3 shows a similar result for f a metric transformation. Here we assume 1im p(d) = 0, hence the doc <:ondition for continuity depends very much on the par- tricular transformation involved. Lemmas 2.4 and 2.5 show a converse of Lemma 2.3. They present conditions under which 1im p (d) = 0. We dao use: this limit later in this Chapter, and in Chapter 3. "L. 25 Lemma 2.6 does not really concern continuity. However it belongs with this series of lemmas, so we include it here. It shows that a metric transformation is trivial (i.e. maps the entire space to a single point) if the domain is connected, and for some 6 > O, p(d) = O for all d < s. This result is used in Chapters 3 and 4. Before proceeding to Lemma 2.1, consider the following example. Example: Let (N,d) be any non-discrete distance space. Define a new distance r on N by O, x = y r(xoy) = d(XoY) + 10 X 7'! Y Define f: (N,d) 4 (N,r) by f(x) = x. Then f is clearly an order isomorphism. However, f is not con- tinuous, for (N,r) is discrete while (N,d) is not. B Thus, not all order transformations need be continuous. Lemma 2.1 shows that many are. Lemma 2.1: If f is an order isomorphism from N1 onto N with scale function p, and N2 is not discrete, 20 then f is uniformly continuous, and lim p(d) = O. dao ' Proof: Let 6 > 0 be given. Since N2 is not discrete, there are f(p), f(q) in N2 with O < d2(f(p),f(q)) < e. Let 6 = dl(p,q) and notice that 5 > O. For any x,y 6 N1 with d1(x,y) < 6 = dl(p,q) we have 26 d2(f(x).f(y)) =p(dl(x.y)) $P(d1(p.q)) =d2(f(p).f(q)) < 6 Hence f is uniformly continuous and lim p(d) = O. D d40 Lemma 2.2: If f is an order isomorphism from N onto N2, then f is bicontinuous if and only if (a) both N1 and N2 are discrete (b) neither Nl nor N2 is discrete. Proof: It is clear that if f is bicontinuous either (a) or (b) holds. If (a) holds then, since an order isomorphism is one- to-one, and every set in each space is open, f is bi- continuous. If (b) holds, the bicontinuity of f follows from the previous Lemma, and the fact that f"1 is also an order isomorphism. D Thus we have established that if N and N are l 2 non-discrete distance spaces, any order isomorphism between them is continuous. Lemma 2.3 appears to be similar to Lemma 2.1, however the condition is on the scale function p (which depends on the metric transformation f). Thus Lemma 2.3 is not nearly as strong as Lemma 2.1. Lemma 2.3: If f is a metric transformation from N into N with scale function p, and lim p(d) = O, 1 2 d-o then f is uniformly continuous. 27 Ppppg: Let 6 > 0 be given. Let 6 > 0 be such that d < 6 = p(d) < e. Let p 6 N1. Then if q is any point Of N1 With dl(Poq) < 6 we have d2(f(p),f(q)) = p(d1(p.q)) < e. The lemma follows. C] The limit 1im p(d) = O which appears in the above d40 Lemmas is important and we wish to study it further. Lemma 2.4, and a consequence of it, Lemma 2.5, will be used in future chapters. These may be considered as a converse to Lemma 2.3. Lemma 2.4: Let f be a continuous metric transfor- mation from N1 into N2 with scale function pp. Suppose there is a p 6 N1, and a number a > 0 such that 0 31b g a implies there is a q 6 N1 with dl(p,q) = b. Then lim p(d) = O. d40 Proof: Let 8 > 0 be given. Since f is continuous, there is a 6 > 0 such that d1(p,q) < 6 ‘implies d2(f(p),f(q)) < e. For every b > O, b < max[a,6}, there is a q 6 N1 with d1(p,q) = b. Hence Nb) = P(d1(p.q))= d2(f(p).f(q)) < e That is, b < maxfa,6} =° POD) < 6:. Hence 1im p(d) = O. [J 640 Lemma 2.5 states Lemma 2.4 in terms of familiar properties of distance spaces. A continuous distance function is needed here, as well as in Lemma 2.6, to con- sider the topological property of connectedness. 28 Lemma 2.5: If (N1,d1) is a distance space with a continuous distance function, containing a connected subset which is neither empty nor a singleton, then any continuous metric transformation f of N1 onto N2. with scale function p, satisfies 1im p(d) = O. d40 Lemma 2.5 follows easily from Lemma 2.4 and the Intermediate Value Theorem. Lemma 2.6 is of a slightly different flavor, but we include it here as it also examines the scale function near 0. This lemma will be used several times in Chapter 4. Notation: For any two sets C and T, C\T is defined tobe [xlxec and xiT]. Lemma 2.6: Let f be a metric transformation from N1 into N2 with scale function p. If for some 6 > O, p(d) = O for all d,O < d < 6. then flc is trivial for any connected component C of N1. Proof: Let C be a connected component of N1 and let pec. Let T= [xEC|p(dl(P.x)) =0}. Note that f(T) = {f(p)}. Let xEET and let y be such that d1(x,y) < so Then d2(f(p).f(y)) d2 0 be a given distance. Let t 6 [a,b] and assume d(y(t),y(b)) 2_r. Then there is an s E (t,b] with d(Y(t).Y(s)) = r- Proof: This is a simple application of the intermediate value theorem. Consider the function g: [t,b] 4 I! defined by g(r) = d(Y(t),y(r)). Then 9 is continuous, since both the distance function and Y are continuous. 9(t) d(v(t).Y(t)) = O 9(b) d(Y(t).Y(b)).2 r 33 By the intermediate value theorem, there is an s E [t,b] with g(s) = r. Also 3 ¥ t for otherwise 0 = g(t) = 9(3) = r, contradicting r > 0. Thus, the lemma is proved. C] Lemma 2.8: Let y = Y([a,b]) be a rectifiable arc in a distance space N with continuous metric d. Let r > 0 be a given number. Then there exists a set 9 =[po,...,pn } in y with Y(a) = P0: Y(b) = P r 0 n r r d(pi-l'pi) = r, 1 g i < nr d (pnr_l. pnr) S r and the set {pi} is normally ordered with respect to y. Proof: Let p0 = y(a). Assume p0,...,pm have been defined. If d(pm,y(b)) g r, 1et pm+1 = Nb). and let nr = m + l. Otherw1se, we Obtain pm+1 as 1n Lemma 2.7 and proceed, as above, to find pm+2. We need now to show that eventually the process ter- minates, that is, that for some m, pm+1 = y(b),. For any m, consider S = [po,...,pm,y(b)}. S is normally ordered with respect to y, hence MY) 2 US) at = Z) d(Pi-l'pi) + d(pmoY(b)) i=1 = i r + d(pmnvfbH i=1 ‘2 mr. 34 Thus m S.£{¥L , so m cannot be arbitrarily large. The lemma is now complete. B One more definition is needed before Theorem 2.9 can be stated. This definition is discussed further after the statement of Theorem 2.9. Definition: The distance space N is said to satisfy the long legged local isosceles property at a point p 6N if and only if there is a number 1(p) > 0 such that for any 6 < l(p), and for any q with d(p,q) < 6 there exists 5 E N’ with d(p,s) = (s,q) = 6. The number h(p) may be a. P >3 q 6 Figure 2.1 Theorem 2.2: .Let f be a metric transformation with scale function p from a metric space (Ml,d1) to a metric space (M2,d2). Assume there is a point p 6 M1 and a number 1 > 0 such that (a) Ml satisfies the long legged local isosceles property at p with x(p) = x. (b) There is a point q 6 M1, with f(p) :1 f(q), and a rectifiable arc ygg M1, joining q to p. (c) The set f({xld1(p,x)<:%3) is pg; uncountably discrete. 35 Then for every t EM the function gt is 1 ‘=fl{x|dl(t.x)<§} 1 bicontinuous. Furthermore f,gt, and g; are all uniformly continuous. The hypotheses of this theorem may seem awxward, but they are easy to apply. Many spaces satisfy the long legged local isosceles property. It will be shown in Chapter 3 that any open subset of a normed linear space (of dimension at least 2) satisfies it at each point. In Chapter 6, it is shown that a smooth hypersurface in En (n 2_3) satisfies it at each of its points. Also, it can be shown that any point of an open subset of either hyper- bolic or spherical n-space (n 2.2) satisfies the long legged local isosceles property. Assumptions (b) and (c) depend on the particular metric transformation, although they seem to be "mild" conditions. Fortunately, in practice, assumptions on the domain and range can often be substituted for these. For example, in Chapter 3 it will be shown that if f: U 9232) v is a metric transformation and U and V are both open subsets of a finite dimensional normed linear space, then (b) and (c) are satisfied. Note that we require both the domain and range of f to be metric spaces. This is unfortunate, and may not be necessary. However we have been unable to prove or to disprove such a theorem without the triangle inequality. The proof of Theorem 2.9 is in 5 parts. The first 'two show the continuity of (at, from which the continuity 36 of f follows easily, the last three that gll is continuous. In part (1) we show that some arbitrarily "small" distances are transformed to "small" distances. It is here that we use (c). In part (2) the long legged local isosceles property and the triangle inequality are used to show that if some arbitrarily small distances are trans- formed to "small" distances, then all "small" distances are transformed to "small" distances. Thus f is continuous. To show 9;} is continuous it is necessary to show "small" distances come from "small" distances. Using the long legged local isosceles property and the rectifiability of y, for each r, O < r < x, a set of points P = qIP IP ...-.9 - .S,p = P O 1 2 nrl nr is constructed with the distance between any two adjacent points being r. Then d2(f(p),f(q)) g (nr-+1)p(r). If lim (nr-tl)p(r) = 0 then f(q) = f(p), contradicting (b). r40 Thus as p(r) becomes "small", nr must become correspond- ingly large, which, it is seen, only occurs if r becomes "small". From this, the continuity of g2} follows. In part (3), the pi's and s are defined, and the relationship between nr and r is studied. In part(4), gt is shown to be one—to-one by verifying that p(r) # O, -1 O < r < 1. Finally, in part (5), the continuity of gt: is established. 37 Notation: Let B(x,r) = [y 6 Mlld1(x,y) < r]. Proof of Theorem 2.9: (1) Let a > 0 be given. Then there is a 6 < k such that p(6) < e. Egoof of (IL: Assume the contrary. That is, assume there is an e > 0 such that for all 0 < 6 < l: p(6) 2,6- Then for any xl-leeB(p,)2‘-) it must be that d2(f(x) .f(y)) = p(d1(x,y));;e. Hence fl is one-to-one, and B (p. x/ 2) f(B(p, %)) is discrete. On the other hand, B(p, %) contains an open subset of the arc y, so is uncountable. Thus f(B(p, %)) is uncountably discrete, contradicting (c), and (1) has been proven. (2) The function f is uniformly continuous. Proof of (2L: Let 6 > 0 be given. By (1), choose 6 > O with 6 < x such that p(6) <'§. Let d < min[6,dl(p,q)] be given. By Lemma 2.7, there is a t E Y such that d1(p,t) = d, and by the long legged local isosceles property of M1 at p, there is an s with d1(p,s) = dl(s,t) = 6. Then p(d) = d2p = 0. Since f(p) # f(q), (assumption (b) of Theorem 2.9) this is a contradiction. Thus, p(r) > O for r < x. If x,y E B(t, %) then dl(x,y) < A so that d2(f(x),f(y)) = p(d1(x,y)) # O, and hence f(x) # f(y). This completes the proof of (4). (5) For each t E M fl is bicontinuous. 1' B (t, 1/2) 39 Proof of (5): Let a be given, 0 < e < 1. Assume that for each 6 > 0, there is an r, e.g r < k such that p(r) < 6. Then by (3), d2 0 such that if r < x and p(r) < 5 then r < e Let t 6 M1 and let gt 5 f) Then by Step B(t,1/2)' 2, gt is one-to-one and uniformly continuous. Let 6 > 0 be given (assume 6 < l). and choose 6 > 0 such that if r < X and P(r) < 6 then r < e If x,y e B(t, %) and d2(f(x),f(y)) < 6 then d1(x,y) < e. l is uniformly continuous, and hence gt This shows that g; is bicontinuous with both 9t and 9;} uniformly continuous. This completes the proof of Theorem 2.9. 13 Corollapy 2.10: If 1(p) = a, then f is bicontinu- OUS . In Sections 3 and 4 of this chapter bicontinuity is usually assumed. Using the above theorem, or Theorem 2.2, this assumption can often be avoided. In Chapters 3 and 4, where metric transformations in normed linear spaces and Euclidean spaces are investigated, the above theorem is applied. 4O 63. The major result of this section is to show that in the class of convex metric spaces, the order of the distances determines the space, up to a similarity. This is the content of Theorem 2.11. Theorem 2.17 is a metric transformation version of this. Before discussing Theorem 2.11 the following definition is needed. Definition: A distance space (N,d) is said to be convex if and only if for any two points p and q of N, there is a segment joining p to q. It should be noted that although this definition fits one's intuitive notion of convexity, and is similar to that normally used in linear spaces, it is not the definition commonly found in metric geometry. The common definition of metric geometry is that for each p and q in N there is an r, r # p or q, such that d(p,r)4—d(r,q) = d(p,q). There are several additional assumptions that can be made on N so that these two definitions are equivalent. Perhaps the best known is that of completeness. For further discussion on this, see [4] sec 14. Theorem 2.11: In the class of convex metric spaces, the order of the distances determines the metric up to a similarity. 41 In other words, if (M1,dl) and (M2,d2) are both convex metric spaces, and f is an order isomorphism from M1 onto M2, then there is a constant D such that d1(p.q) = D-d2(f(p).f(q)). A metric transformation version of this was probably originally due to Wilson, for he states in [39], without proof, that a bicontinuous metric transformation with non-zero finite spread, between convex metric spaces, is a similarity. This is the content of Theorem 2.17, although the assumption that the spread is non-zero and finite is not made there. Theorem 2.11 is a consequence of Theorem 2.17. In [2], Beals, Krantz, and Tversxy show Theorem 2.11 with the added assumption of completeness. In [23] Lew proves a stronger version of Theorem 2.11. He insists on neither completeness nor as strong a version of convexity as we use here. He uses "pseudo-convexity". Definition: A metric space (M,d) is said to be pseudo—convex if for any two points p and q, and x in [0,1], and any 6 > 0 there exists u such that d(pou) g e+xd(p.q) and d(q.u) : e+<1-x)d(p.q) It seems not to be known whether the completion of a pseudo—convex space is convex. 42 In the work of Beals, Krantz and Tversxy, and that of Lew, the primary interest is that of existence. They show that, for a certain class of distance spaces, each space is order isomorphic to one and only one convex metric space. Here we are considering only the uniqueness question and, like Wilson, use metric rather than order transfor- mations. Our methods could be adapted to pseudo-convex spaces, giving Lew's result, although the proofs would become more involved. Leading up to Theorem 2.17 we first show that if f :M -+M is a bicontinuous metric transformation, and l 2 y is an arc in M1, then L(f(y)) = D-z(y), where D is the spread of the transformation (see Theorem 2.14) . From this it follows that the image of a geodesic arc is a geodesic arc (Corollary 2.15) and then, in convex spaces, that the image of a segment is a segment, (theorem 2.17). Theorem 2.16 shows that, under very general conditions, the spread of a metric transformation is a non-zero finite number. We begin.with Lemma 2.12, which shows that arc length can be calculated by considering only "small" distances along the arc. Note that the triangle inequality is necessary. 43 Definition: For any normally ordered subset P = [p0,pl,...,pm} of an arc y, the number mesh(P) is defined to be max[d(pi_1,pi)|1 g.i g m}. i Lemma 2.12: Let y be any rectifiable arc of a metric space. Then for each n > 0 there is a 5 > 0 such that any normally ordered subset P of y with mesh(P) < 5 satisfies g(P) > g(y) - n. Proof: See Blumenthal [4], page 61, Lemma 24.1. I: Remark: If y is a non-rectifiable arc, then Lemma 2.12 may be stated as follows: For each N > 0, there is a 5 > 0 such that any normally ordered subset P of Y with mesh(P) < 5 satisfies 1(P) > N. Blumenthal does not prove this, but the proof is essentially the same as the proof of Lemma 2.12. Corollary 2.13: If rn 4 O and Pn is a normally ordered subset of an arc y with mesh(Pn) g rn then 4(Y) = lim £(Pn) . 114' It is not assumed in this corollary that y is rectifiable. 44 Theorem 2.14 (Wilson [40]): Let M1 and M2 be metric spaces, f: M1 4 M2 a bicontinuous metric trans- formation with finite non-zero spread D, and y a rectifiable arc in M1. Then f(y) is a rectifiable arc in M2 and 1(f(y)) = Dog(y). If y is a non- rectifiable arc, then L(f(y)) = a. Ppgpfi: Note that f(y) is an arc in M2, the equation of the arc being f 0y(t), a g.t g_b. We will prove the case 1(y) < a. The case 1(y) = a is essen- tially the same, using the above remark, rather than Lemma 2.12. By Corollary 2.13 1(y) = lim 4(Pn), where Pn n49 is any sequence of normally ordered subsets of y with lim mesh(Pn) = O. n-fi Let c > 0 be given. As D = lim (d) , there is d40 a 5 such that for d < 5, D - e < (d) < D + e, or (D-e)d < p(d) < (D+e)d. For any normally ordered subset P = [po,...,pm] of y with mesh(P) < 5, f(P) is a normally ordered subset of f(y) and 1 (f (P)) = d2 (f(pi-1)' f (‘31)) '5' Ma 1 hence m m (D- 6,51 d1 (Pi-1'91) s L (f (P)) g (D+ 6351 51(Pi-1'Pi) or (D-e)z(P) guflPH g (D+€)£(P) Let Pn be any sequence of normally ordered subsets of Y with mesh(Pn) 4 0. Then f(Pn) is a sequence of normally ordered subsets of f(y), and as f is uniformly continuous on the compact set y, 1im mesh(f(Pn)) = O. 11-” Hence by Corollary 2.12 £(f(y)) = lim L(f(Pn)). 11-90 For n such that mesh Pn < 6 we have (D-e)2(Pn) S “f(PnH S (D+e)£(Pn) Letting n 4 a, we get (D-EHM S 2(f(v)) g (D+e)£(Y) As e was arbitrary, it must be that £(f(y)) = D°£(y). Thus if 2(v) < a, then L(f(y)) = D°£(y). As has been said, the case £(Y) = a is much the same. In this case £(f(y)) = a. I] Recall from Chapter 1 that a geodesic arc joining p to q has the shortest length of any arc joining p to q. 46 Corollapy 2.15: If y = y([a,b]) is a geodesic arc in M1 and f: Ml‘QflggiMz is a bicontinuous metric transformation with finite non-zero spread D, then f(y) is a geodesic arc in M and £(f(y)) = D-L(y). 2’ Proof: That £(f(y)) = D-L(y) is the content of Theorem 2.14. Let 0- be an arbitrary arc in M2, joining f(y(a)) to f(y(b)). Then by Theorem 2.14, 1(0) = D-.¢,(f-l (0)). Hence inf 1(0) = inf DUE-1(0)) 2 D1. (y) . However 0' O £(f(y)) = D-£(y), showing f(y) is a geodesic arc in M2. (3 Before continuing the study of metric transformations of arcs we show (Theorem 2.16) that the assumption in Theorem 2.14, that f has finite non-zero spread, is in many cases unnecessary. Theorem 2.16 may be considered as a converse of Theorem 2.14. Theorem 2.16: If the metric spaces M1 and M2 each contain a rectifiable arc of length greater than zero, onto 1 then the spread of f is a non-zero finite number. and f: M > M2 is a bicontinuous metric transformation, Proof: Let y= y([a,b]) EM and O = 0([c,d]) EM 1 be rectifiable arcs, and let p be the scale function 2 of f. Let {ri} be any sequence of numbers such that P(r.) 1im r1 = O and lim -—-l- = DIg m. Since y is connected, i4a 14¢ i Corollary 2.5 guarantees that lim p(ri) = 0. With this 47 fact, using notation as in Lemma 2.8, we have by Corollary 2.13, L(f(y)) = lim nr p(ri) and L(y) = 1im nr.ri° 14° 1 14¢ 1 Then n e(r-) p(r.) r. 1 D=1im r1 =lim—n'l'T—=LéfT(-)xu. i4a i i4o rii‘ Y Since £(f(y)) is bounded from below by d2(f(v(a)),f(y(b))) and £(Y) is a positive real number, we see that D > 0. Similarly we show that r. -l lim_1___._£_(.f_1£zll=l>0. iaaphfi) 4(0) D Thus 0 < D < a. Since D =-&%%6¥LL is independent of [ri], it must be the case that lim Eéfl = D, proving r40 the theorem . Cl Remark: At this point it should be noted that the assumption in Theorem 2.14 that f be a bicontinuous metric transformation with finite non-zero spread is satisfied for f an order isomorphism between two "reasonable" metric spaces. Using Lemma 2.2 and Theorem 2.16 we see that if both the domain and range of an order isomorphism f contain a rectifiable arc, then f is a bicontinuous metric transformation with finite non-zero spread. With the above remark in mind, Theorem 2.11 follows easily from Theorem 2.17. 48 Theorem 2.17: If M1 and M2 are convex metric spaces, and f is a bicontinuous metric transformation from Ml onto M then f is a similarity. The constant 2’ of similarity is the spread of f. Proof: From their respective definitions, it is immediately clear that a segment is a geodesic arc, and any two geodesic arcs, joining the same points, have the same length. Let p and q be arbitrary points in M1. Since M1 and M are convex, there are segments y and o: joining 2 p to q and f(p) to f(q), with lengths dl(p,q) and d2(f(p),f(q)) respectively. By Theorem.2;16, f has finite spread D. By Corollary 2415, f(y) is a geodesic arc of length D-L(y). Furthermore, since f(y) is geodesic, it has the same length as On Thus d2(f(P)of(q)) = £(f(Y)) = D°Z(Y) = D'dl(poq) Since p and q were arbitrary, the theorem is proved. l3 §4. The major result of this section, Theorem ZMZO, concerns the curvature of arcs in general metric spaces. We first define what is meant by curvature of arcs in general metric spaces, and relate this to the more common definition of curvature in Euclidean and Riemannian spaces. Definitions: Let y = y([a,b]) be an arc in a metric space (M,d). If q = y(s) and r = y(t) we define 49 Ly(q,r) = 1(y([s,t])). That is, Ly(q,r) is the arc length of y from q to r. For p,q,r on r, if the following (non-negative) limit exists, it is called the Haantjes-Finsler curvature of y at the point p: def 1y(q.r)«-d(q.r) 2 . (Yip) = 11m 4'. KB q.r4p z:(q.r) This definition of curvature was first introduced by Finsler in his thesis of 1918 [13] and was studied extensively by Haantjes [15]. We are going to study the relationship between KH(y,p) and KH(f(Y),f(p)), where f is a metric transformation. The curvature KH lends itself to this study for we have a nice grasp on the rela- tionships of LY(q,r) to Lf(Y)(f(q),f(r)) and d1(q,r) to d2(f(q),f(r)). In Euclidean space,curvature is usually defined as follows: Definition: If y(s) is a curve in Euclidean n-space, where the parameter 3 represents arc length along the curve, the curvature of y at a point y(so) is defined to be IY”(sO)I. This is referred to as the classical curvature of y at the point y(so). More generally, classical curvature is defined for arcs in Riemannian spaces by means of the Frenet formulae. We refer the reader to a book on Riemann geometry, such as Spivak [37], for such definitions. 50 We now quote some theorems which relate these concepts of curvature. Theorem 2.18: (Haantjes) If an arc in a Riemannian space has defining equations with continuous differentials of order 3, and if the classical curvature exists at a point on the arc, then the curvature KH exists at that point, and the two curvatures are equal. Proof: See [15], Theorem 5. For arcs in Euclidean space we can combine another theorem of Haantjes ([15], Theorem 8) and a theorem of Egervary and Alexits ([11], Theorem 4.2) to obtain the following result. Theorem 2.12: If y is a curve in a Euclidean space, and the classical curvature exists at p, then it is equal to A (q.p)-d(q.p) lim 4! v 3 q4p zY(q.p) The limit in the above theorem certainly exists if the Haantjes-Finsler curvature of Y at p exists, in which case the two are equal. This limit could be used as a definition of curvature: however we will stay with the more common KH. Theorem 2.20 remains true if KH is replaced by the limit of Theorem 2.19. In Theorem 2.20 it is assumed that f is a metric transformation with spread 1. There is essentially no loss 51 of generality since,if f: M1 4 M2 has finite non-zero spread D, we define a metric r on by M2 __ l —— —— ~ r(p,q) ='B d2(p,q), p,q 6 M2, and let f: (Ml,dl) 4 (M2,r) be defined by f(p) = f(p). It is now easily seen that f is a metric transform with spread l and that (M2,d2) and (M2,r) are similar. Theorem 2.20: Let M1 and M2 be two metric spaces, f a bicontinuous metric transformation of M1 onto M2, with spread 1. Then (a) If y is any arc in M1' p E Y and KH(y,p) and KH(f(y),f(p)) both exist, then lim d g'd exists and d4O d (1) K§ - Kfi.f) -d; V c M 2 (a) m 2.n (b) If m>n, then forany subset B of U we have g(B n U) = g(B n U) n v and this set contains no open subset of M2. (c) If m = n and U is open in M1 then g(U) is open in M2. Proof: If m < m, then because the topologies on any two finite dimensional N.L.S. of the same dimension are equivalent, it is necessary only to prove these for M1 = En, M = Em (Euclidean spaces). Thus (c) follows directly from 2 Theorem 3.5. (a) Assume m < n. Then m < a and we need only consider M1 = En, and M2 = Em. Consider Em as a subspace of En. Then 9 : B(p,r) 4.Em’c En, and by Theorem 3.5, g(B(p,r)) is open in En. However as g(B(p,r)) : Em this cannot be true. Thus m 2_n, and (a) has been proved. (b) That g(B n U) = g(B n U) n V for any subset B of U follows directly from the fact that g is bicontinuous. Assume m > n. If g(B n U) contains an open subset V’ of M2, let M c M be a linear subspace of M2, of finite 2 . I dimension greater than n, such that M n V # ¢. Then 62 is bicontinuous from M n V’ c M into M1. hence by part (a) dim M g.n. This contradicts the choice of M. Thus g(B n U) contains no open subsets of M2. [3 To conclude this section three lemmas are presented. The second, Lemma 3.8 is a commonly used property of convex quadrilaterals. Lemma 3.7 is presented only to prove Lemma 3.8. Lemma 3.9 has surely been shown by others, although we have not seen a proof of it. It does not extend to 3 dimensions. the that Lemmas 3.7 and 3.8, while stated in terms of N.L.S., are in more general metric spaces. Definition: Three vectors a1,a2, and a3 are said to satisfy the triangle equality if and only if for some permutation (i,j,k) of (1,2,3), ”ai-ajH-tHaj-a I==Hai‘-a k, k” ° Lemma 3.7: Let M1 be a N.L.S. with a strictly convex unit ball. Then any three vectors are collinear if and only if they satisfy the triangle equality. Proof: If three vectors are collinear it is easy to check that they satisfy the triangle equality. To prove the converse assume the vectors are p,q, and r and that Hp-—r”-+Hr-—q” = Hp-—q”. We show that r E [Poq]. If any two of p,q, or r are equal it is easy to see that r 6 [p,q]. Assume they are distinct. Then none of HP'-qH. Hp-—rH, “r -q” is zero, and p-r p-r r- r- _ 2-3 'ip-qi ' IIP-rII+'IP-qn 'nr-qil ' HP-qll ' 63 Also l_wtg%=lm-qH4m-rn= r- p-q Hp-qH Hp-q“ Thus, letting x = N. we have - r r .. - + l.- = - 1”%:?n ( M W7fi%[ Hgtgn . "r r "' Certa1n1y 0 < A < l, and the three vectors [[5711]? fir, H§€33H all lie on the boundary of the unit ball. By the definition of strict convexity, they must all be equal. In particular Rearranging this last equation we obtain r = xq4—(l-—x)p, concluding the proof of Lemma 3.7. [3 Lemma 3.8: In a 2-dimensional N.L.S. 1et a,b,c,d be the vertices of a convex quadrilateral, given in a cyclic order around the edges of the quadrilateral. Then (a) Ha -b|l + H(2 -dll S Ma ‘CH + Nb 'dll - That is, the lengths of two opposite sides add up to no more than the sum of the lengths of the two diagonals. (b) If in addition a,b,c, and d are distinct, not all collinear, and the unit ball is strictly convex then the inequality above is strict. 64 a( d Figure 3.1 Proof: The proof rests on the fact that the diagonals [a,c] and [b,d] intersect at a point p which lies in the interior, or on the boundary of,the quadrilateral abcd. Ha “H! = Ha “PH + HP '0“ Nb ‘dH = ”P “P" + HP ‘6‘” - (1) However ”3 '1’” S H3 ‘PH + HP ‘19” ”C ‘5” _<. ”5 ‘PH + HP ‘CH . (2) Adding the two inequalities of (2) and combining with (l) we obtain Ha-bH+Hc-du5;Ha-CH+Hb-dfl To show part (b) of the lemma it is only necessary to show that one of the inequalities of (2) is strict. If not, Lemma 3.7 shows that a,b, and p are collinear ppg_that c,d, and p are collinear. Since p 5 ac n bd we can readily conclude that either not all the points are distinct, or all are 65 collinear, contradicting the hypothesis of (b). Thus the inequality must be strict. [3 Lemma 3.9: In any two dimensional N.L.S. with strictly convex unit ball, there is at most one point equidistant from three distinct points. gpppg: Let p,q,r,c1, and c2 be such that ”p-ciH = ”q-ci” = ”r-ciH, i = 1,2. Also assume cl 7! c2, and p,q,r are distinct. If p,q, and r are collinear, the strict convexity of the unit ball is violated. Consider Figure 2. If either c1 or c2 lies in one of the closed regions 1,2, or 3 the convexity of the unit sphere is violated. The line clc2 can intersect at most two of the segments (p,q), (q,r), or (p,r). Assume it does not intersect (p,q). Then either pqclc2 or pqczc1 forms a convex quadrilateral. Assume the latter. The points p,q,c1,c2 are distinct and not collinear, so by Figure 3.2 66 Lemma 3.8(b) HP-CzHi'Hq-Cl” > IIP-C1H+ [lg-C2” However Hp-c2H = ”q-c2H and Hp-—cl” = Hq-—c1H, giving a contradiction. Thus c1 = c2. In §2. We are now prepared to investigate the continuity of metric transformations between subsets of N3L.S. Theorem 2.9 gave sufficient conditions for a metric transformation to be continuous. This section proceeds by determining when these conditions are satisfied. We recall the definition of the long legged local isosceles property. Definition: The metric space M has the lppg. legged local isosceles property at P if and only if if there is a number x(p) > 0 such that for any 5 < A(p) and for any q with d(p,q) < 5 there exists 5 e M with d(p,s) = d(s,q) = 5. The number x(p) may be taken to be c. There are four results in this section. These are Lemmas 3.10, 3.11, 3.13 and Theorem 3.12. Lemma 3.10 shows that any point p of an open subset of a N.L.S. has the long legged local isosceles property. Lemmas 3.11 and 3.13 then show sufficient conditions for the continuity of a metric transformation between subsets of N.L.S. Lemma 3.13 differs from Lemma 3.11 in that it considers only finite dimensional N.L.S., obtaining stronger results in this case. Theorem 3.12 is a theorem of Vogt, which follows easily from our work. 67 Lemma 3.10: Let U be a subset of a N.L.S. M, with dim M 2.2. Assume that p and x > O are such that B(p,x)s:U. Then U has the long legged local isosceles property at p, and k(P) can be taken to be x. Proof: Let 5 < x be given, and let q be given with Hp-q” < 25. If S(p.6) nS(q,5) = (6 then S(q,5) = (B(p,5) nS(q.o)) u ([xld(p.x) > a} ns(q.5)) Since both members of the above union are non-empty (for example H-tq is in the first member, while - “(p -q,) +q is in the second), both members are Open subsets of S(q,5), and S(q,5) is connected, we have a contradiction. Thus, S(p.5) ns(q.6) :4 ¢. Let seS(p.5) nS(q.5). Then d(p.s) = d(q,s) = 5. Since d(p,s) = 5 < x then seB(p,)‘) and hence s EU. [:1 Remark: In the above lemma, one could replace B(p,x) by any subset B, peBcU, such that B is an isometric image of an open ball of radius x in a N.L.S. of dimension 2. Then it would be necessary to verify only the case dim M = 2 and the result would follow immediately. Lemma 3.10 is used in proving Lemma 3.11, for it allows Theorem 2.9, on the continuity of metric transformation to be used. Example 2 at the end of this chapter shows the necessity of the assumption that dim M1 2.2. 68 There are undoubtedly many conditions which force a metric transformation from a subset U : M1 to M2 to be continuous. Lemma 3.11 presents two such conditions, the proofs of which use Theorem 2.9. Lemma 3.11: Let f :U-4M2 be a metric transformation, Uch B(p,x) : U. , dim M1 2.2. Let p and x > 0 be such that (a) If f(B(p,x/2)) is not discrete then le(t,x/2)r1U is bicontinuous for each t E U, and f is uniformly continuous. (b) If M2 is separable, then le(t,x/2)IWU is either bicontinuous for each t E U, or is trivial for each t 6 U. If le(t,A/2)(1U is trivial for each t e U, then flc is trivial for each connected component C of U. Proof: .As B(p,x)s:U, Lemma 3.10 shows that p e U has the long-legged local isoceles property (as a point of the metric space U) with x(p) = A. (a) If f(p) = f(q) for all q 6 B(p,x/2) then f(B(p,x/2)) is a singleton, hence discrete. Thus there is a q E B(p,x/2) with f(p) # f(q) and letting y = [p,q] c U. Theorem 2.9 shows that fiB(t,x/2) is bicontinuous for all t 6 U, and that f is uniformly continuous. 69 (b) Assume that there is a q E B(p,1/2) with f(p) ¥ f(q). If f([p.q]) is uncountable then, because M2 is separable, f([p,q]) contains a limit point of f([p,q]), (in fact a condensation element). Hence f(B(p,x/2)) is not discrete. Letting y = [p,q] Theorem 2.9 now shows that le(t,x/2) is bicontinuous for all t e U. If f(p) = f(q) for all q 6 B(p,x/2) then certainly p(d) = O for o < d < k. The proof follows by Lemma 2.6. [3 At this point the following theorem, due to Vogt [38], is easy to show. Theorem 3.12; (Vogt) Let M1 and M2 > M2 be a metric transformation. be N.L.S., and onto 1 Then f is an affine similarity. dimMl 2.2. Let f :M 3599;: By Lemma 3.10, the origin has the long legged local isosceles property, with x = 9. Since M2 = f(Ml) = f(B(O,%)) is not discrete, Theorem 2.9 shows that f is bicontinuous. Theorem 2.17 then shows that f is a similarity, and by the theorem of Mankiewicz (Theorem 3.1) we condlude that f is an affine similarity. [3 Remark: Corollary 3.15 generalizes this to the case onto f :M, > V : M2, V open in M2. It is shown there that 1 necessarily V = M2. Although our proof of Theorem 3.12 is somewhat different than Vogt's, our work on the continuity of metric transformations 70 (Theorem 2.9) was motivated by his work in [38] *where he proves the above theorem. Mankiewicz makes use of techniques from this same paper in proving Theorem 3.1. Lemma 3.13 strengthens Lemma 3.11 when separable and finite dimensional N.L.S. are considered. It is here that the Invariance of Domain Theorem (Theorem 3.5) is used. Lemma 3.13: Let M1 and M be separable N3L.S. and onto 1 2 formation. Assume that there exists p,q 6M1 and rl,r2 > O 2 > V c M dimMl 2,2. Let f :U c M be a metric trans- suchthat B(p,rl) :U and B(f(CI)or2) CV- Then f|B(t,rl/2)()U is bicontinuous for all t E U and f is uniformly continuous . In particular f'B(p r /2) is bicontinuous and nontrivial. ' 1 If in addition, dim M1 = n < a, then dim M2 = n. Proof: As M is separable, and B(p,rl) c U, Lemma 2 3.11 shows le(t,rl/2)r1U is either trivial for each t e U or bicontinuous for each t 6 U. Since M1 is separable, and a susbspace of a separable metric space is separable, then U is separable. Let T be a countable dense subset of U. Let c. be the countable collection of subsets of U of the form B(t,r1/4) n U, t 6 T. Note 6 covers U . If f]B(t,r1/2)r1U is tr1v1al for each t 6 T, then V = U f(B(t,rl/4)r1U) would consist of a countable set of tET points. Since B(f(q),r2) : V, this is impossible. Thus it 71 follows that le(t,rl/2)rfiU is bicontinuous for each t e U. In particular, le(p,rl/2) is bicontinuous, and hence non- trivial. If dimM1 = n < a, then Corollary 3.6(a) shows dim M2 Z'n. Assume dimM2 > n. Then by Corollary 3.6(b) f(B(t.r1/4) mm = f(B(t.r1/4) nU) nv and this set contains no open subsets of M Hence 2. B(f(q).r2) [u f(B(t.r1/4) nUH nB V is an order transformation, is f a similarity? In this chapter, U and V are subsets of normed linear spaces, called M1 and M2 respectively, and transformations, f. are invariably metric rather than order transformations. The class c. just referred to, may be taken to be some suitable collection of subsets of M1. For example, one of the hypotheses of Theorem 3.14 is that the domain and range be open, hence a suitable choice for a. would be all open subsets of M1. Each of Theorems 3.14, 3.16, and 3.18 and Corollary 3.19 involve some type of openness hypothesis on the domain and/or range of f. This is done in order to use methods developed in Chapter 2 for convex sets. In addition, some further hypotheses is needed. For example, in Corollary 3.19, M1 and M2 are assumed to have the same finite dimension. 73 Note that these results could be considered as gener- alizations of Mankiewicz's Theorem (Theorem 3.1) to metric transformations. Definition: A function is said to be Open if and only if it maps Open sets onto Open sets. Theorem 3.14: Let M1 and M2 be N.L.S., dim M1 2’2. Let f be an open metric transformation, from an open connected set U of M1 onto an open set V of M2. Then f is an affine similarity. Proof: Let p, and x > 0 be such that B(p,)[) :U. Then by hypothesis, f(B(p,x)) is open, hence not discrete. So, by Lemma 3.ll(a), f]B(t l/2)()U is bicontinuous for each teEU, and f is uniformly continuous. Let t be an arbitrary point of U. By hypothesis, f(B(t,x/2MWU) is open in V, and hence in M2. Let r2 be such that B(f(t),r2)s:f(B(t,x/2)(WU). As f is continuous f-1(B(f(t),r2)) is open in U, and hence in M1. Let r1 be such that 74 1 B(t.r1) :f‘ Vs:M dim M1 2.2, V a 2' non-empty open subset of M2, and f is a metric transformation, then f is an affine similarity and V = M2. 75 Proof: As in the proof of Theorem 3.12, f is necessarily bicontinuous, hence is open. So by Theorem 3.14, f is an affine similarity. Thus, for each x 6 M1, f(x) = T(x)4-d where T is a linear transformation and d 6 M2. To show V = M ’1et y 6 M2 and p 6 M1. Then because 20 V is open there is a x # 0 such that (1-x)f(p)4-xy E V. Hence there is a q 6 M1 such that f(q) = (li-x)f(p)-+xy. Then = f(q) -(1-}.)f(p) y 1 = N9.) - (1 ->.)T(L)+d l k = f(q- (1 ->.)p) l and hence y 6 f(Ml) = V, so V = M2. [3 Theorem 3.16: Let M1 and M be NyL.S., 2 g dim M1 < a. 2 Let f be a metric transformation, f :U onto > V, U an open connected subset of M1, and VCMZ. (a) If V contains an open subset of M2 and M2 is separable, then f is an affine similarity and dim M1 = dim M2. (b) If dim M1 = dim M2, then f is either trival or an affine similarity. 76 Proof: If V contains an open subset of M then 2’ Lemma 3.13 shows that dim M1 = dim M2, and f is non- trivial. Thus if we assume dim M1 = dim M2, and f is non-trivial, and then show that f is an affine similarity both (a) and (b) will have been proven. This is what is now done. Assume dim M1 = dim M2 and f non-trivial. Let p and x be such that B(p,x)s:U. Then by Lemma 3.11(b) (because M2 is a finite dimensional vector space, it is separable) fiB(t,x/2)t1U 13 either bicontinuous for each t e U or trivial for each t e U. Furthermore, Lemma 3.11(b) shows that if f'B(t.x/2)r1U is triv1al for each t 6 U, then flc is trivial for any connected component C of U. Since U is assumed to be connected, this would imply that f is trivial, contradicting the assumption that f is non- triv1a1. Thus we may assume that fiB(t,x/2)11U is bicontinuous for each t E U. As B(t,x/2)r1U is an open subset of M1, and by the assumption dimM1 = dim M2, Corollary 3.6(c) shows that f(B(t,x/2)r1W) is open in M2, for any t E U, and any open subset W :U. 'Ihus, given an arbitrary open set W :U, f(W) = [J f(B(t,x/2)11W) is an open subset of M2, so f tEU is an Open mapping. Theorem 3.14 now shows that f is an affine similarity. 0 Note that Theorem 3.16 (a) generalizes Mankiewicz's theorem to metric transformations between open subsets of finite dimensional N.L.S. 77 We have tried, without success, to establish the results of Theorem 3.16 for the case in which both M1 and M2 are separable. Theorem 3.17 summarizes the progress we have made. Theorem 3.17: Let M1 and M2 dili 22. Let Uch, VcM be separable N.L.S., 2 be open, and let f :U 9222) V be a metric transformation with scale function p. Then there is a x > 0 such that le(t,x/2)r1U is bicontinuous for all t e U, and an r > O and q 6 U such that f‘B(q,r) is an affine Similarity, and le(t,r)¢1U is a similarity for each t e U. Remark: If it could be shown that f(B(t,r)) is open in M2 for all t, it would follow that f]B(t,r) is an affine similarity and Theorem 3.1 would show that f is an affine isometry. Proof of Theorem 3.17: Let p and x be such that B(p,x)s:U. Then Lemma 3.13 shows that le(t,x/2)¢1U is bicontinuous for all t. As in the proof of Lemma 3.13, for some b e‘u f(B(b,x/2)r1U) must contain an open subset of M2. Let f(a), and r1 be such that B(f(a),r1) : f(B(b,x/2)r1U). Now le(b,x/2)r1U is bicontinuous, hence w = (leOLX/z) DU)-I(B(f(a),r1)) is open in B(b,A/2), and hence open in M1. Let B(q,r)s:W, and g e le(q,r)' Then g is a bicontinuous metric transformation of B(q,r). The set g(B(q,r)) is open in M2, for g(B(q,r)) = is f(B(q.r)) :f(W) = B(f(a)or1) and f]B(b,)\/2) nU bicontinuous. It then follows from Theorem 3.14 that In (’3 78 g a f'B(q,r) is an affine Similarity, hence p(d) = D-d for some D > O, and for all d < 2r so f]B(t r) nU' is a Similarity for all t. [3 When working in a N.L.S. with a strictly algebraically convex unit ball (see the definition at the beginning of this chapter) we are able to obtain results which are stronger than those of Theorem 3.14 and 3.16. These results, contained in Theorem 3.18 and Corollary 3.19, might be suprising since the ‘ set Uch ' is not assumed to be connected. Theorem 3.18: Let Ml strictly convex unit ball. Let Us:M1 be any set with non- , dimMl 2_2 be a N.L.S. with a empty interior. Let f be a metric transformation, f :U 4f(U) cMz. Assume for some p E U, x > O that B(p,x) : U, and f(B(p,x/2)) is open in M2. Then f is an affine similarity. More important than this theorem may be the following corollary. Corollary 3.19: Let M1. 2 ngim M1 < a, be a N.L.S. with a strictly convex unit ball. Let U c M1 be any non- empty set which contains an open subset of M1. Let f :U-+M be a metric transformation, 2 (a) If dimMl = dim M2 then either f is an affine Similarity or f1C is trivial for each connected component C of U. 79 (b) If f(U) contains an open subset of M2, and M2 is separable,then f is an affine similarity. Proof of Corollary 3.19: (a) Assume dim M1 = dim M2. Note that dim M2 is finite, hence M2 is separable. Let p 6 U and x > 0 be such that B(p,x) :‘U. Then by Lemma 3.11(b), either flc is trivial for each connected component of C of U or f|B(t,x/2)11U is bicontinuous for each t 6 U. In the latter case, le(p,x/2) is bicontinuous, hence by Corollary 3.6(c), f(B(p,x/2)) is open in M2. Part (a) now follows from Theorem 3.18. (b) If f(U) contains an open subset of M2, and M2 is separable, Lemma 3.13 shows that n = m and f]B(p,x/2) is bicontinuous. Hence f(B(p,x/2)) is open in M2 (Corollary 3.6(c)). Then (b) follows from Theorem 3.18. [3 Progfiof_Theorem 3:18: Let g I le(p,x/2)' Then 9 is a metric transformation, 9 :B(p,x/2)-+f(B(p,x/2)). Since f(B(p,x/2)) is open, it cannot be discrete. Hence Lemma 3.11(a) shows that g is bicontinuous, and so is an open function. From this it follows by Theorem 3.14 that g is an affine similarity. Thus g = A‘B(P:l/2) where A is an affine transformation and also a Similarity of M1 onto M2. Let a > 0 be such that HA(x)«-A(y)” o”x-—y“, for all x,y 6 M1' Let q 6 U and consider f(q) and A(q). Let d > 0 be such that S(q,d) n B(p,A/2) # O. Let T be any (two 8O dimensional) plane in M2 containing f(q) and A(q), and intersecting g(S(q,d)r1B(p,x/2)). Since A is an affine Simularity from MI ontg M2 and g E A|B(Pol/2) we have 9(S(q.d) nB(p.%)) A(S(q.d) nB(p.§)) S(A H = ”f(xz) -f(q)|l = “f(x3) -f(q)u . Since 7r is the translation ofa two dimensional N.L.S. with strictly convex unit ball and A(xi) = f(xi) for each i we can apply Lemma 3.9 to the equations (1) and (2) to see A(q) = f(q) . Now q being an arbitrary point of U, we have f _ AlU' and f iS thus an affine similarity. D Corollary 3.19 is our final result showing metric transformations between subsets of N.L.S. are similarities. , Theorems 3.14, 3.16, 3.18 and Corollary 3.19 would seem to give some validity to the statement "the order of the distances determines the set up to a similarity". On the other hand, we have lOOked at specialized situations and undoubtedly there are many order transformations between subsets of N.L.S. which are not similarities. Unfortunately the only type of example we have been able to find has a disconnected domain - Example 1. Example 1 shows that the strict convexity of the unit ball is necessary for Theorem 3.18, and that the connectedness condition of Theorem 3.14 is also necessary. Example 1: Let M1 = 1:. That is, M is the set of 1 all ordered pairs (x,y) with ”(x,y)” = max[x,y]. Let 82 S1 = {(x,y) |H(X.y)H < l} (I) ll 2 {(xoy) I H (xay) - (4.0) H < 1} For any real number a > 0, define f by f(x.y) = (x,y). (x,y) 6 S1 f(XIY) (x+aIY) I (XIY) 6 82 Then f is a metric transformation, with scale function p(d) = {34a 3 g g , but f is certainly not a Similarity. E] Throughout this chapter it has been assumed dimM1 2_2. This was used to obtain the continuity of the metric trans- formation. In the case of order transformations, as has been mentioned, Lemma 2.2 rather than Theorem 2.9 can be used to show continuity. In this case all of the theorems could be changed to include dim M1 = 1. To Show the assumption dim M1 2_2 is necessary for metric transformations, consider the following, due to Vogt [38]. Example 2: (Vogt) Consider Hi, the real numbers, as a vector space over the field 0, the rational numbers. Let A be a basis for this vector space, and assume 1 6 A. For a 6 A, define f by f(a) = {:a': : i and extend f to I ]R by linearity. Now considering H! as a N.L.S., (of dimension 1) the function f : IR-olR is a metric transformation because [f(X) -f(y)| = [fix-y)! = [affix-371)] = [f(IX-YIH and the scale function of f is p(d) = [f(d)[. 83 On the other hand f is not a similarity for p(l) = 1, while p([a]) = 2]a], a e A, a # 1. More generally, define a metric transformation f : :IR-tM, M any N.L.S., by defining f(a), a e A arbitrary and then extending f by linearity to IR (treating ]R as a vector space over Q). As above ”f(x) 'fmll = “fix-y)” = ”affix-YUM = “f(lx-YIHI- This may be stated more simply by noting that f is a group homomorphism from ]R into M, although possibly the above description may yield more insight. Thus many non- continuous metric transformations of fit into any N.L.S. can easily be constructed. Even metric transformations onto finite dimensional N3L.S. can be constructed as the cardinality of a basis of a finite dimensional N.L.S., treated as a vector space over the rationals, is the same as the cardinality of Hz, treated as a vector space over the rationals. In Chapter 4 all metric transformations, both continuous and . . n . non-continuous from E: into E are characterized. [3 This concludes our work on metric and order transformations of general normed linear spaces. In Chapter 4 we shall consider the problem.further when working with subsets of En. . CHAPTER 4 n ORDER AND METRIC TRANSFORMATIONS IN E The original interest of M.D.S. Theorists was in order transformations into En. Typically their concern was with a finite space, which we regard as a distance Space, and with the possibility of finding a subset of a low dimensional Euclidean space order isomorphic to it. Thus, order embeddings into En have assumed a prominent role in M.D.S. theory. A number of interesting questions on the existence of order embeddings into En have been examined by Holman, Kelly and others. We have made some efforts to extend these without much success. For completeness, we summarize these results in Chapter 6. Some of our earlier results have immediate implications for embeddings into En. The most interesting of these may be Corollary 3.19 which shows that if s is any subset of En, containing a ball, then any order or metric transformation of S into En (in fact, into any n-dimensional normed linear Space) is a similarity. This result is extended in Section 2 of this chapter. 84 85 It was mentioned in Chapter 1 that von-Neumann and Schoenberg [29] have characterized all continuous metric transforms of it into Hilbert space. These are the so called pgpgy_curves, whose associated scale functions are given by 2 (**) p2(d) = c d2 +23 Ai sin 2 Kid, ci'Ai'Ki being constant. Much of this chapter depends on this characterization. The chapter is divided into four sections. Section 1 is bacxground material for the rest of the chapter. Here we briefly discuss the concept of a metric basis. These are often implicitly used in metric geometry, although rarely mentioned by name. In Section 2, a theorem of Schoenberg is extended. In [32] Schoenberg showed that if f : Em 4En, m 2 2 is a continuous metric transformation, then f is a similarity. Vogt [38] showed that the assumption of continuity could be dropped. we Show the domain need not be all of Em, giving two different types of subsets of Em for which the theorem is still valid. Section 3 contains the characterization by von-Neumann and Schoenberg of the continuous metric transformations of JR into En. This characterization is extended to the non- continuous case. We also Show in this section that any metric transformation of an interval of JR into En can be uniquely extended to a metric transformation of IR into En . 86 This is important, in Section 2, where it is used to Show that (**) is a characterization of the scale function of a metric transformation of an interval, as well as of the real line. Although parts of Section 3 are used in Section 2 it seems appropriate to present the results of Section 2 first. The main result of Section 3 (the characterization (**)) is not new, and our contributions are not surprising. In addition, we suspect that all of these results are contained somewhere in the literature. Section 4 of this chapter is quite separate from the other sections. It considers metric and order transformations of the unit ball in En (and in other inner product spaces) into normed linear spaces. This will be discussed further when we come to it. 61. In this section we define and state some theorems concerning metric bases. This material is used in subsequent sections of this chapter. Definition: A subset B of a distance Space N is said to be a metric basis of N if and only if each point p of N is uniquely determined by the set [d(p,b) [besB]. Metric bases abound in the classical Spaces. A subset of En, Sn, or Hn is a metric basis if and only if it does not lie in an (n-l)-flat. Such a set is said to span En, Sn, or Hn respectively. 87 n A metric basis of En, S or HD must contain at least (n4-l) points, and any metric basis (spanning set) of En, Sn or Hn contains a subset of exactly (n4-1) points which is itself a metric basis. We refer the reader to Blumenthal PI] for the details of these statements. Metric bases have appeared earlier in this thesis, although they were not called such. The proof of Theorem 3.18 partly consisted of showing that any open subset of a normed linear space with strictly convex unit ball is a metric basis for that space. Metric bases also appear briefly in Chapter 5. The following two lemmas are important properties of metric bases. These are stated for En, but they are also valid for Sn and Hn. Lemma 4.1 is a commonly used property of En. It is often referred to as the property of "free mobility". Definition: An isometry of En onto itself is called a motion of En. Lemma 4.1: If f : S cEn4En is an isometry, then f can be extended to a motion If of En. If S contains a metric basis B, then the extension is unique. Proof: See Blumenthal [44, Sec 38]. Lemma 4.2 shows that a metric basis is not always needed to use the same type of arguments, provided more information is available. 88 Lemma 4.2: Let Em be an m-flat in En and let BcEm be ametric basis of Em. If perm and qun, and d(p,b) =d(q,b) for all bEB, then p=q. Proof: See Blumenthal [44, Sec 40]. This essentially says that points of Em are determined by a metric basis of Em, even within the larger space En. Corollary 4.3: If f is a motion of En which maps a metric basis of Em into Em, then f(Em) = Em. The following lemma is used in Section 2. Lemma 4.4: Let f :s c EV4EV. Let Eu be a u-flat in EV, and Us:SnEu be such that flU is an isometry. Let BcU be a metric basis of E‘1 and let pES be such that leLJ{ is an isometry. Then flULJ{p] is an p) isometry. In particular, d(p,q) = d(f(p), f(q)) for all qGEU. Proof: By Lemma 4.1, and flu can be extended f _ IBu[p] to motions 'f and 'f respectively of EV. The set f(B) is a metric basis of both the u-flats f(Eu) and fXEu), from which it follows that "f(Eu) = is“). Both f] u and :f] u extend le to an isometry from E E Eu to f(Eu). By Lemma 4.1, there is only one such extension. 73" . [Eu Hence 'fl Eu For qEUch, f(q) = ?(q) = f(q). Also. f(P) = ?(p), Hence, flULJ[p] s f‘U(J[p] is an isometry. D 62. In this section we extend the result of Schoenberg [32] which Shows that any continuous metric transformation of Em into En, is necessarily a similarity. The result depends on the following characterization, by von—Neumann and Schoenberg [29]. of the scale functions associated with screw curves in En. v (**) p2(d) = c2d2+ Z A? sinzk.d . 1 1 i=1 where v=n'2'1, n odd, v=§ for n even and c= O, and v=r-l—§—2- for n evenand C710. An examination of Schoenberg's proof reveals that in fact he proves the following. (Recall that a trivial metric transformation is one which maps the entire set to a single point.) Theorem 4.5: Let Ss:Em contain a circle C and a line L. Then any continuous metric transformation of S into En is either a similarity or trivial. Theorems 4.6 and 4.7, which follow, extend this result even further. 90 Theorem 4.6: If Ss:Em contains a circle C, an unbounded connected subset T, and a line segment L with 1.513 then any continuous metric transformation of S into En is either trivial or a Similarity. Theorem 4.7: Let S :Em, m 2 2 be a connected set with non-empty interior. Then any metric transformation of S into En is either trivial or is a similarity. The subject of continuity needs comment. Theorem 3.11 (b) shows that any metric transformation from an open subset of a normed linear space (of dimension greater than one) into a separable normed linear space is either continuous or trivial. Thus the assumption of continuity is not necessary in Schoenberg's version of Theorem 4.5. This is proved by Vogt in [39]. Theorem 4.7 includes the assumption of an open subset of Em, so Lemma 3.11 (b) shows continuity there. On the other hand we are not guaranteed an open subset of F.m in the set S of Theorem 4.6, so Lemma 3.11 cannot be applied We conjecture that continuity need not be assumed in this case, but follows from the remaining hypotheses. However we have not been able to Show this. The proofs of Theorem 4.6 and 4.7 are adaptations of Schoenberg's proof of Theorem 4.5. For this reason we present Schoenberg's proof of Theorem 4.5. 91 Proof of Theorem 4.5: Let f be a continuous metric transformation of S into En. We proceed by examining the scale function, p, associated with f. First consider f restricted to 5. Since 1 is isometric to ER, f(t) is a screw line, thus p must satisfy (**). That is, s (**) p2(d) = c2d2+ z A? sinzk.d, i=1 '- 1 for all d and some non-negative values of c, Ai' and Ki. Figure 4.1 Next consider f restricted to C. For any pEC, 1et 0(p) be the central angle, measured between the radius with endpoint p, and some fixed radius. For any two points p1 and p2 of C, with 01 = 0(pl), 02 = Osz) we have O'-O d(p1,p2) 2r sin ‘ 2 1| and I02"01l p( 2r Sin -——17——- ) = p(d(p1.pz)) d(f(pl).f(p2)) 92 l0 W! Since p( 2r sin -—21r—£— ) depends only on [02-01|, the mapping cp : IR-oEm defined by cp(o(p) + 2k1T) = f(p) for each k, is a continuous metric transformation of fit into En with scale function add) = p([2r sin gl). Hence by (**) ~ ~ v (1) p2(|2r sin 3]) = p2(o) = czoz-t 2) Bi sinzhio i=1 Since C' is bounded, f(C) and hence g(nu are also bounded, so it must be the case that ‘3 = 0. Thus p(er sin $1) = v 23132 . i sinzhio. Let d = |2r sin g], and setting the i=1 expressions (**) and (l) for p equal, we get v p s E B? sinzh.a = 4r2c2 sin2 3+ Z) A? sin2(2k.r sin %) i=1 1 1 i=1 1 1 It follows from this (see the appendix at the end of this chapter) that either Ai or ki is O for each i. Thus (**) reduces to p2(d) = c2d2, or simply p(d) = cd, c 2.0. Since (**) is true for any distance d, f is either a similarity. or is trivial. I] To adapt this idea to Theorems 4.6 and 4.7, several problems arise. The most serious of these would seem to be the substitution of a line segment L for a line 1 in the hypotheses. It is not clear that the scale function associated with a continuous metric transformation of a line segment, rather than a line, has the form (**). It is shown in Section 3 of this chapter (Theorem 4.12) that any metric transformation of an open interval can be extended, uniquely, to a metric 93 transformation of the entire real line. Assuming the validity of this result, it follows that the scale function associated with a continuous metric transformation of a line segment has the form (**). It can then be shown in the same manner as in the proof of Theorem 4.5 that if Ss:En contains a circle and a line segment, then any metric transformation of S into En satisfies p(d) = cd, for any distance d which occurs as the distance between two points on the circle, 9; two points on the line segment. In Schoenberg's case (Theorem 4.5) this Shows p(d) = cd for 311_ d (hence that f is a similarity) however it does not do so for us. To overcome this problem, extra hypotheses are needed, (Example 1 at the end of this section shows that some extra conditions are indeed needed), such as the connectedness condition in each theorem. Before proceeding to the proofs of Theorems 4.6 and 4.7, we Show the following lemma. Lemma 4.8: Let S be a connected set and 1, a line in E:m such that Snz contains an interval U of 1. If f :SaEn is a metric transformation, with scale function p, and fiU is an isometry, then d(f(p).f(q)) = d(p,q) for all p68. qu Proof: Let qO be an arbitrary point of U. Since U is open in 1, there are points ql and q2 of U such that the segment joining ql to q2 is in U, and 94 d(qolql) = d(qofllz) . Let r = d(qoiql) = d(qooqz) do = sup[d|d = d(P.qO) a PE 5] d1 = sup[d’|p(d) = d, for all d < d’,d = d(p,q),p,q e S]. In other words, d1 is the largest distance such that p(d) = d for all d in [0,d 1). Figure 4.2 Since fiU is an isometry we have d1 2.2r > O and d0 2.r > 0, hence min[do,d1} > 0. Let d2 be an arbitrary distance less than min[d0,d1]. Let p be an arbitrary point such that d(p,qo) = d2. There is at least one such point because S is connected, and d2 < do. Now since d2 < d1, 95 there are two points b1 and b2 of U with d(p,bi) < <11 and hence with p(d(p,bi)) = d(p,bi). Let M be an Euclidean space with EmcM, EncM. Then we can apply Lemma 4.4, with f,p,S and U as above, Eu = 1, V E = M, and B = [bl,b to show that 2} p(d(p.q)) = d(f(p),f(q)) = d(p,q) for all q€.U. . . )/2 2 In particular, p(d) = d for all d in [0, dzi-r ). (See Figure 4.2. Apply the law of cosines, noting that one of the angles p q0 q2 or p q0 q1 is between w/2 and w.) As d2 is an arbitrary distance less than min[do,d1], it follows that p(d) = d for all d in [O,\/Qmin[do,d1])2-tr2). If d = min[do,d1} this is a contradiction. Therefore 1 d0 = m1n[do,dl] and hence p(d(p,qo)) = d(p,qo) for all pegs. Since qO is an arbitrary point of U, the lemma is proved . [3 Corollary 4.9: Let S be a connected set in Em, and m assume U is an open subset of a line 1 in E with U c S. Let f :SeEn be a metric transformation such that flU is a similarity with P(d(q1:q2)) = Cd(q10q2) I C > 0: <11qu 6 U Then p(d(p.q)) = cd(p.q) for all p e s, q 6 U 96 Ppppg: Define ‘f by f(p) = % f(p). Then it is easily seen that 'f is a metric transformation and ‘flU is an isometry. Let '5 be the scale function associated with 'f. Then certainly B(d) = 51:- p(d). It follows from Lemma 4.8 that p(d(p,q)) = d(p,q) for any pes, qu. Hence p(d(p,q)) = cd(p,q) for any pes, qu. [3 We are now prepared to prove Theorems 4.6 and 4.7. Theorem 4.6: If S :Em contains a circle C, an unbounded connected subset T, and a line segment L with I.:Tb then any continuous metric transformation of S into n O O O I O Q I I E is either tr1v1al or is a Similarity. Proof: Let f be a continuous metric transformation of S into En, with scale function p. Using Theorem 4.12 fiL can be extended to a screw curve, hence p must have the form 2 22 r 2 2 (**) p (d) =cd + Z) A. sin k.d . 1 1 i=1 for any distance d that can be written as d = d(ql,q2), where q1,q2 e L. Using C, the proof of Theorem 4.5 can be imitated to show that p(d) = cd, c 2_O, for any distance d that can be written as d = d(q1,q2), q1,q2 E L. (Recall that the proof of Theorem 4.5 depended upon (**), the existence of a circle C in S, and the continuity of f.) 97 If c = 0, then p(d) = O for d less than the length of L and, by Lemma 2.6, f is trivial. If <:#'O Corollary 4 .9 shows that p (d(p,q)) = cd(p,q) for any p e T, q e L, which defines p(d) for any distance d. Hence f is a similarity. [:1 Remark: When applying the above theorem, Lemma 3.11(b) could well guarantee that the continuity assumption is satisfied. Theorem 4.7: Let S :Em, m 2 2 be a connected set with non-empty interior. Then any metric transformation of S o n o e c o o I o 0 into E is either a Similarity or is triv1al. Proof: Let f be a metric transformation of S into En, with scale function p. That f is continuous follows from Lemma 3.11 (b). Since m 2_2, and S contains an interior point of Em, S contains both a line segment and a circle. Therefore, as in the proof of Theorem 4.6, p(d) = cd for small d. If c = 0 Lemma 2.6 shows that f is trivial. Now consider the case c > 0. Let f(p) = % f(p). and let 3' be the scale function corresponding to ‘f Clearly, N p = é p, and hence ‘p(d) = d for small d. Thus there is an open ball B of S such that ‘le is an isometry. Noting that for q E B, there is an open segment L with q 6 L c B, Lemma 4.8 shows that d(f(p),f(q)) = p(d(p.q)) = d(p.q) for every p65. qu 98 Now ‘f(B) is an isometric image in En of an open set of Em, so it must be that m gDn. Let M1 be any n-dimensional Euclidean space containing Em, and let M2 be the m-flat of En which contains fXB). See Figure 4.3. Em S f /”“9 M1 Figure 4.3 Recall that an open subset of Em is a metric basis of Em, and hence B is a metric basis of Em. The isometry ij can be extended to an isometry f- : Em 4M , Let p 6 S. 2. For any q 6 B, it has been shown that d(p,q) = d(f(p) ,f(q)) . Also d(p,q) = d(f(p),f(q)) = d(f(p),f(q)) Hence d(f(p),f(q)) = d(f(p),f(q)) for all q 6 B Lemma 4.2 now shows f(p) = f(p). 99 In particular, f(p) 6 M2. Thus f':S : Em-oMz. The spaces En and M2 are normed linear spaces of the same finite dimension, so Corollary 3.19 (a) shows that f, is a similarity. (In fact ‘f is an isometry.) From this it follows that f is a similarity. [3 Example 4.1: To Show the connectedness conditions of Theorems 4.6 and 4.7 cannot be eliminated (although they can surely be modified) consider the following. Let Dn' n any integer, be an open disc of radius one in 3 O the xy-plane, centered at (4n,0). Let f : [J Dn-+E be n=-° defined by f(X.y) = (x,y.n) for (x,y) 6 Dn If d(p,q) = d(p’,q’) where p 6 Dn' q 6 Dma P' E Dnzo q’ 6 Dm’ then [n-m] = [n’-m’] and dz(f 0, let Cr be the circle of radius r, centered at the origin. Let (r,e) be the polar co-ordinates of a point p on the circle. Define f :Cr-4 E4 by f(p) = (/3r2 - 2r4 cos 9, ./3r2 - 2r4 sin 9, J52: r2 cos 29, ./-2- rzsin 29) . It is easy to see that f(Cr) is a metric transformation of [.1 Cr' The scale function associated with flc , called pr, is r given by 101 pr(d) = (3r2--2r4')sin2(sin-l £%)4-% r4sin2(2 Sin-l £%) Since this last expression is independent of r, it has been shown that for pl,q1 6 Cr and p2.q2 6 Cr with 1 2 d(p1.q1) = d(p2.q2> = d. then prl(d) = pr2(d). We have now shown that to prove Theorem 4.7, it is not enough to consider a set of circles and distances which occur between two points of the same circle. [3 A more interesting example would be a set in Em containing two circles of different radii and.a metric transformation of that set into En which is not a similarity. We conjecture that no such example exists. 63. The work in Section 2 of this chapter is based on the characterization (**) of the scale function of a continuous metric transformation of it into En. The transformation. relative to a suitable co-ordinate system, is that which takes teEIi to t, A Sink t,...,A COSk t, A sink t, ct) v v v v 1 1 (A1 cos kl where Ai'xi' and c are constants. If f(nu spans En, then v = 3, c = O for n even, and v = 2434, c # O for n odd. 102 In E2 this says the only continuous metric transforms of IR, (screw curves) that span the plane, are circles. Since an order transformation is necessarily bicontinuous, then there are no order transforms of Hi spanning E2. In E3 the screw curves are necessarily circular helixes. As in Example 4 Chapter 1, these may be order transforms of IR. This behaviour is typical of the even and odd dimensional cases. In even dimensions the screw curves spanning the space are bounded simple closed curves lying on a sphere. Hence there are no order transforms of n: spanning even dimensional spaces. The simple closed curves referred to above are sometimes called "trigonometric moment" curves. In odd dimensions the screw curves spanning the Space are unbounded and may be thought of as generalizations of the circular helix in E3. Such a curve lies on a "spherical cylinder" and its projection onto the base of the cylinder is one of the trigonometric moment curves referred to above. The tangents to the curve make a fixed angle with its axis as in E3. The von-Neuman-Schoenberg study shows much more, being set in general Hilbert spaces. For example it is Shown that bounded screw curves in V' must lie on a sphere. In En screw curves are, of course, rectifiable, but such is not the case in N3 For example the order transform of IR. given by the square root function is an example of a homeomorphic image of the line no sub-arc of which is rectifiable and it was Shown in [32] that this is a screw curve in ya Von-Neuman 103 and Schoenberg characterize those screw curves in y' which are rectifiable. Their proofs involve rather sophisticated analytic techniques applicable to the Hilbert space setting. Since M.D.S. theorists are more concerned with finite dimensions and these characterization are finding their way into the M.D.S. literature we thought a strictly finite-dimensional derivation of the characterization in En might be both simpler and more revealing. We have another reason for presenting a proof in finite dimensional space. Vogt [38] gives an intriguing example, presented and generalized in Chapter 3 (Section 3), Showing non-continuous metric transformations of HI. It is not hard to Show that this example is typical of all metric transformations of Hi into n:. That is, any metric transformation of fit into El is a group homomorphism, followed by a translation. A metric transformation of E: into E2 that is not of this type is the following. Let a : 1R 4 ]R be a group homomorphism. Define f : ]R-+E2 by f(t) = (cos e(t), Sin g(t)). Certainly this need not be continuous, and it is not hard to Show f is a metric transformation. In Theorem 4.10 all metric transformations of Ti into En (be they continuous or not) are charac- terized. We see that they are "made up" from the above types of metric transformations. 104 To assist the reader, we list the various symbols used in the remainder of this section. To be consistent with the literature, norm notation shall be used for distance here. Notation for Theorems 4.10, 4.11, 4.12: S,S.,a ,b - real numbers 3 s s x,y,xj,yj,v1j,v2j,v3j,w - real vectors (The definition of a real vector is introduced in the proof.) z,zj - complex vectors Vj,W.,W - subspaces of Cn (DU US,U. - unitary transformations of Cn TS,T. - affine isometries UNLJ I - the identity transformation Note: Us (or T3) does not represent the transformation U (or T) raised to the power 5, but is one symbol. We write it in this way because it is shown that Usour = Us+r (TsoTr = Ts+r). Let V1,...,Vm be vectors in Cn (vl,v2) - the standard inner product in en [v1,...,vm] - the subspace of en spanned by v1....,vm If Vccn is a subspace, we denote by vi- the set [zecn](z,v) =0 for all vEV]. "VI” = ~/_—(V1"’1) 105 Definition: A unitary transformation Cn.»cn (or En-aEn) is a function U such that U(O) = O and Hzl-22H==HU(zl)-—U(22)H for all 6 Cn (or En). 21,22 The following is a list of elementary properties of unitary transformations. These are all easy to show, and can be found in linear algebra or functional analysis reference booxs. (See for example [1.], [18]). Let U be a unitary transformation. U1. U is a linear transformation of en (or En). Thus U is an affine isometry. _ n n U2. (21,22) - (U(zl),U(22)) for all z ,2 5C (or E). 1 2 U3. The absolute value of any eigenvalue of U is 1. U4. U-l exists. U5. (21'U(22)) = (u‘1(zl).zz). U6. If V is a subspace of Cn (or En) and U(V) = V, then U(v‘) = v‘. U7. If antbi is an eigenvalue of U, then 51%E{==a-bi 1 is an eigenvalue of U- Theorem 4.10: Let fun-.3“ be a metric transformation with scale function p, such that f(fiU spans En. Then there are mutually orthogonal two dimensional subspaces V1, . . . ,Vm of En, an (n - 2m) -dimensiona1 subspace W of En with W ij for all j, vectors 106 lj' 23., and v3]. in Vj, With Vlj'LVZj and "Vle = ”vsz = l, a vector w eW, group homomorphisms V V ej :lR-oR/ZTr, 1 gj gm a group homomorphism g : IR4W, and constants Aj such that F . fj (t) Aj((cos ej(t))v1j+ (Sin ej(t))v2j) +v 33' (*) < fw(t) g(t) +w where fj(t) is the projection of f(t) into Vj, and fw(t) is the projection of f(t) into W. Conversely, any function f : lRaEn of the form (*) is a metric transformation of JR into En. Proof of Converse: We Show that if f satisfies (*) for sane set of normal subspaces V1, . ..,Vn and W of En, then f is a metric transformation. In this case we have ”f(tl) ‘f(t2)”2 = jg], 4A3g sin2(ej(t1) -ei(t2)) + H£W(tl) ‘£w(t2) "2 = jgl 4A; S"’inz(ej(i=1-2.112)) + ”g(tl 't2)l!2 = jg 4a? sinzu. 91-(lt12-tzl))+”:g(‘tl -t2])l[2 = jg]. 4A3? Sin2(ej(‘t12-t2‘)) + H9Ht1 ’t2])|l2 - This shows f is a metric transformation with Scale function p(d) satisfying 5 2 - 2 .2 d p (d) — .23 4A3. em ej(2)+'[9(d)[[2 . c: 3=1 107 We now provide some motivation for the proof of this theorem by informally discussing the situation in E3. 3 Any motion of E can be described (see [6], Chapter 7) as either (A) A rotation about a line, and a translation along that line. This line is called the axis of rotation. (B) A rbtation about a line, and a reflection through a plane perpendicular to that line. (C) A reflection through a plane, and a translation along a line in that plane. We are given a metric transformation f of the real line, with [f(t) , t 51R] Spanning E3. We can define an isometry T3 of the set [f(t) [1:513] by Ts(f(t)) = f(t+s). It is not hard to verify that this is an isometry. Extend this to a motion of E3, also called Ts. As [f(t) [teIR] spans E3, this extension will be unique (Theorem 4.1). Using these properties of T3, it can be shown that (1) Ts+reT3oTr III 6 0T In particular TS e TS/ons/2 for all s, from which we can conclude that T3 is not of type (B) or type (C), hence is of type (A). Next, using (1), it can be shown that the axis of rotation of each TS must be the same. Thus, if Ts, Tr, and T3."r have rotation angles e(s), e(r), and e(s-tr) respectively, then e(s)-+e(r) = e(s-tr) (modulo 2w). It also follows 108 that if TS has a translation along this axis of a distance g(s), Tr of a distance g(r), and T8...: a distance g(s+—r) then g(s)4—g(r) = g(s-tr) (where g(s), g(r), g(s4—r), can be either positive or negative, depending on the direction of the translation). Definition: A set of motions [T8 1 s 6 1R] of En s+r satisfying TseaTr = T is called a one parameter subgroup p§.motions. It follows immediately that any two members of a one parameter subgroup of motions commute. Comment on Notation: Note that T3 is a motion, and does not represent "T raised to the power 3". When we wish n to use exponentiation, and we shall, we write (Ts) . Thus n s) =‘Ti<>Ts 0 ..... 0T3; It is easy to see that if f n times (T [T8 ] s 6 IR] is a one parameter subgroup of motions, then n . (Ts) e Tn 3. As TS and Tr commute. it follows that n n n n (125”) = (TS .145) = (TS) (Tr) . Proof of Theorem 4.10: The proof consists of eight steps. Step 1. For each s 6 1R, define a motion T5 of En as follows. Let Ts(f(t)) f(ti—s). AS f is a metric trans- formation it follows that ”Ts(f(tl)) -Ts(f(t2))” = []f(tl+s) -f(t2+s)H = p([tZ-tll). 109 Hence TS defines an isometry of f(nU into En. By hypothesis f(flU spans En, hence TS can be uniquely extended to a motion of En (Theorem 4.1). Because of this uniqueness, and the fact that T3 oTr(f(t)) = f(t+s+r) = Ts+r(f(t)), it follows that T3 4Tr s Ts+r. Thus [T5 l S 61R] forms a one parameter subgroup of motions of En. The function Ts(x)-TS(O) is a norm preserving map from En onto En, which takes 0 to 0, hence is a unitary transformation (by definition). Thus Ts(x) = Us(x)4-TS(O) where U3 is a unitary transformation. By U1, US is a I O s O O I linear transformation, hence T is an affine isometry. Remark: Step 1 shows that every metric transformation f of it into En can be written as f(s) = Ts(f(0)), s O 0 where T is a one parameter subgroup of motions of En. Step 2. Consider tn, the complexification of En. Extend Us to a unitary transformation on Cn, also called 3 U , by Us(x+ iy) = Us(x) + iUS (y) where x and y are real vectors. (That is to say, x and y lie in the natural embedding of En into Cn. See [1] for a discussion of complexification.) Note that this is the -1 unique extension of Us to a linear map of Tn. As (Us) exists (by U4) it follows that Us(y) = 0 if and only if y = 0. 110 Hence if z = xi-iy (x,y real) then Us(z) = Us(x4-iy) = Us(x)4-iUs(y) ir real if and only if z is real, that is y = O. This property of US will be used frequently Extend TS to CD by Ts(z) = Us(z) +Ts(0) Note that this is the unique extension of Ts to en such that Ts(z) is an affine isometry of On. Also, because 3 TS(O) is real, and U (z) is real if and only if z is real, then Ts(z) is real if and only if z is real. Step 3. Because the extension of Ts to Cn is unique it follows that T 0'1‘ = T = T <>T showing TS is a one parameter subgroup of motions of en. New TS 0Tr(z) Ts(Ur(z) +Tr(0)) Us (Ur(z) +US (f(a)) + T3 (0) and 3+]? Ts+r(z) = Us”(z) +T (0) Hence, combining the above equations S-t s-tr (U r-USUr)z U3(Tr(0))+TS(O)-T (0) constant As this is true for all z, the constant is O, and s-tr S'Fr (1) USUr = U and T (0) = US(Tr(O)) +T5(0). Similarly 111 UrUs = Us+r and Ts+r(0) = Ur(TS(0)) +Tr(0) Thus (2) UsUr = UrUS and Us(Tr(0)) +TS(0) = Us(Ts(0)) +Tr(0) or (3) (I -Ur)'rs(o) +Us(Tr(0)) = TWO). where I is the identity transformation. Step 4. We now wish to “simultaneously diagonalize" the transformations Us,s 61R. That is, we wish to find a basis B of en such that the matrix of Us in B is diagonal, for all S. We have enough information to apply standard theorems of linear algebra, however we need to be quite specific in the choosing of the basis vectors so we present some details of the usual arguments. Equation (2) and a standard theorem of linear algebra ([2], P 206) Show that the transformations Us,s 61R possess a common eigenvector which we call x+ iy, where x and y are real. Assume |lx+iyu=‘/s2-. Let the eigenvalue of US corresponding to x+ iy be as + ibs . For any 5 61R, Us (x) + iUs (y) = Us (x + iy) (as + ibs) (X + iy) asx-bsy4-i(bsx4-asy) As observed earlier, Us (x) is real if and only if x is real, hence it follows that 112 Us (x) = asx -bsy Us (y) = be+ asy Now U3 (x - iy) = Us(x) - iUs (y) = (as -bsi) (x - iy), shOWing that x -iy is an eigenvector for all Us, the corresponding eigenvalue being as -bsi. Case I. If bs = O for all S, 1et W1 = [x+iy, x-iy]. It is easy to verify that Wl = [x,y] . Note that W1 may be either one or two dimensional, but in any case, it has a real basis. Clearly US (W1) = Us([x+ iy, x -iy]) = W1 hence by S J. = J. Case II. Assume that bS #0, for some S 51R. We 1 1 now Show (x+ iy) .L (x -iy) and that x .Ly. We have 5‘1 (4) (x+iy, U (x-iy)) = (a +b i)(x+iy, x-iy) 51 51 By U70 5"1 -1 (5) ((U ) (x+‘iy), x-iy) = (as --b8 i) (x+iy, x-iy) 1 1 By US, the left hand sides of (4) and (5) are equal, hence Subtracting (5) from (4) yields 0 = 2b i(x+iy, x-iy) 31 As bS 7‘0, we have (x+iy, x-iy) =0. Then 1 o = (x+iy, x-iy) = HxH2-”y||2+21(x,y) 113 From which it follows that (x,y) = O and ”x” = Hy”. Also . 2 2 2 . 2 = ||X+1YH = ”XI! +HYH +21 Combining these, we see that ”X” = ”Y” = 1 Let x1 = x, y1 = y and let Vl be the space spanned by the orthonormal vectors x1 and y1. For any 9, S S . . U (v1) = U ([x14-iy1, X1-1y1]) = v1, hence by U6, Us(Vi) = vi. For any 3, the restriction of Us to Wi in Case I, or Vi in Case II is a unitary transformation hence we can continue the decomposition of On, in the same manner, by lOOking at this restriction. Thus it follows that n C = V1@V2® . . .®Vm®WlWZG-) . . .@W K The subspace Vj has the orthogonal basis {in'iyj' xj-iyj} and the real orthonormal basis [xj.yj}. For some 3, say S. I s o the eigenvalue of U 3 associated with xji-iyj, call it Sj Sj SJ spanned by two real vectors (which may be dependent) so it too a 4-ib is not real, ie b # O. The subspace Wj is has a real basis. If W = Wle...@WK. then ‘W has a real basis. Note, it may be that Cn = VrOVze...®Vfi or on = w. s _ s s _ 3 Step 5. ILet Uj - U lvj, UW — U ‘W’ For each 3 let S _ s s s T (0) — Tl(0) + +Tm(0) +TW(0) 114 where T§(O) evj, Tam) 6W. AS Vj, j = l,...,m and W have real bases, and Ts(0) is real, it is not hard to conclude that T;(O), j = l,...,m and T;(O) are themselves real. Define T;(z) = U§(z) +T:.'(O) and 1:;(2) = age) +T‘:(0) If 2 = zl+ ...+zm+zw, zievi, zWEW, it follows that 125(2) = )3 (Us(zj) +T§<0H +Us(zw) ”3(0) = E (Ugmj) +T3°r g, but J. Lawrence [22] has shown that odd sided 122 regular polygons span only even dimensional spaces. His reasoning employs transformation ideas similar to those in our proof. Our early reflection on the metric determinancy problem led us to conjecture that regular polygons with a large number of vertices would probably be planar but,as the above discussion shows,this was rather naive. Even the ultimate regular convex planar figure, the circle, has order isomorphs spanning arbitrarily high-dimensional Euclidean Spaces. §4. Most of the results of this thesis, up to this point, have been of the following type. The function f is a metric transformation, whose domain contains a convex subset. Then f is shown to be a similarity by first showing that the range also contains a convex subset (compare Theorems 2.16, 3.16. 3.18, 4.7). In this section we Shall consider order transformations of the unit sphere of an inner product space. The unit sphere of such a Space is certainly not convex, so it seems we have managed to move away from the convexity conditions. However, as the surface metric of the sphere of an inner product space, and the nonm metric, are related by an order transformation (see Lemma 4.14), convexity again plays a role. Notice that we have been talking in terms of order transformations here. The main theorem of this section, Theorem 4.15, is stated, and proved only for order transformations. This theorem shows that an order transformation from the unit sphere of an inner product space onto the unit Sphere of a 123 normed linear space must be a similarity, and the N.L.S. must be an inner product space. The theorem is, at least for finite dimensions greater than two, true for metric, rather than order transformations. The proof of this generalization is outlined after the proof of Theorem 4.15. The proof of Theorem 4.15 relies on the following characterization of inner product Spaces, due to Senechalle [34]. Theorem 4.13 (Senechalle): Let M be a normed linear space with unit sphere S. Then M is an inner product Space if and only if there is a function F such that for each p and q in S FWP-qw =HP+qH Proof: See [34]. The following lemma is an easy consequence of some of our work in Chapter 2. It is an interesting result about metric transformations, although it certainly is not surprising. The assumption of bicontinuity is not necessary for dim E1 2_3, as Theorem 2.9 can be applied. However, for our purposes, the following suffices. Lemma 4.14: Let S1 and S2 be the unit spheres of inner onto 1 >32 be a bicontinuous metric transformation. Then f is an isometry. product spaces E1 and E2 respectively. Let f :S 124 gpppg: Let the metric given by the inner product on S1 and 52 be d1 and d2 respectively, and let the intrinsic metrics be di and dé respectively. (The intrinsic distance between x and y being the length of the shortest arc of the great circle joining x to y.) Note that (Sl,di) and (Sz,dé) are convex metric spaces. Let g1 :(Sl,dl)-4(Sl,di) and g2 :(Sz,d2)-4(Sz,dé) be given by gl(x) = x, g2(x) = x. Both 91 and g2 are order transformations with scale function p, where p(r) = 2 sin '1 g . Thus the function g2fgi1 is a bicontinuous metric transformation between the convex metric spaces (Sl,di and (Sz,d’ . Theorem 2.17 then shows that ngg;1 is a similarity, and it follows that indeed it is an isometry. Hence. f is an isometry. [3 Before proceeding to the proof of Theorem 4.15, recall from Chapter 1 that an order transformation has a one-to-one strictly increasing scale function. Hence the scale function and the order transformation are both invertible. See Chapter 1, §2. Theorem 4.15: Let (E,d) be an inner product space with unit sphere S. Let M be a normed linear space, with unit sphere U. If f :s °nt° > U is an order transfonmation, then f is an isometry, and M is an inner product space. Proof: Let H'H be the norm in M and let p be the scale function for f. Then p is one-to-one. First it is shown that U is strictly convex, and then that antipodal points of 125 S are mapped by f to antipodal points of U. Since 2 is the maximum distance between points of S, f preserves the order of the distances, and 2 is the maximum distance between points of U, it follows that p(2) = 2. Let xe; S. The scale function p of f is one to one, and -x is the only point a distance 2 from x, so it follows that f(-x) is the only point a distance p(2) = 2 from f(x). This shows that f(-x) = -f(x). Let iyeu. Then ”35+?” = Us" (5) H = p(d(f‘1(§).f’l(-§)>) ... p(d(f‘1(§).-f‘1<§))> .... p(fl.a2(f‘1(§) ,f-1(§))) = 95/45. (p'1(u'£-§H))2) f‘1(§) ...1 — _ _ —l - '1 _ f (-y) - f (y) 2 7 f (Y) Figure 4.4 126 Senechalle's Theorem, Theorem 4.13, now shows that M is an inner product space. Lemma 4.14 now shows that f is an isometry . C] Corollary 4.16: Let (E,d) be an inner product space with unit sphere S. Let M be a N.L.S. with unit sphere U and dim M = dim E < m. If f :S-+U is an order trans- formation, then f is an isometry, and M is an inner product space. 2599;: Let dim M = dim E = n. Note that U is homeomorphic to S. Any proper open subset of S is homeomorphic to an open subset of En-l' and hence by the Invariance of Domain Theorem (Theorem 3.5), is mapped by f onto an Open subset of U. Thus f(S) is open in U. On the otherhand f(S) is a compact subset of U, hence (as U is a Hausdorf space) f(S) is closed in U. Since U is connected, it follows that f(S) = U3 U Corollary 4.17: If dim E = dim M = 2, and f :S-oM is an order transformation then M is an inner product space (ie the Euclidean plane) and f(S) is similar to S. Proof: Fix a point p of S. Then for x 6 S we have the first configuration shown in Figure 4.5. 127 f (-X) f(p) " f <-p) f(x) ’ Figure 4.5 ‘ Since f is an order transformation, it is easy to see that the rectangle (-x)px(-p) is transformed into a parallelogram, whose diagonals have equal length. These diagonals bisect each other at a point 0, the midpoint of [f(-p),f(p)]. Thus Hf(p)-—OH = ”f(x)‘-O”, and hence f(S) lies on the circle [y] ”y-—OH = %p(2)}, so the corollary follows from Corollary 4.16. D As indicated earlier, Theorem 4.15 remains valid if the assumption of "order transformation" is replaced by that of "metric transformation", and 3 g_dim E < a. The proof of this is only outlined here. Theorem 4.15(a): Let M be a N.L.S. with unit sphere U. onto Let S be the unit sphere of En, n 2_3. If f :S > U is a metric transformation, then f is an isometry, and M is an :n-dimensional inner product space. 128 Outline of Proof: That it is necessary for n to be at least 3 follows by noting that a circle is a screw curve, and can be mapped onto itself by a discontinuous metric transformation. For example, elt-.es3(t) where o is any group homomorphism of TR. (Example 2 of Chapter 3 discussed this type of metric transformation.) The proof that f is an isometry is the same as that of Theorem 4.15 once the following three facts are established. The function f is bicontinuous, f(-x) = -f(x), and the scale function p is one-to-one. We shall now informally discuss the proofs of these in terms of the three dimensional case. The proofs remain virtually the same for higher finite dimensions. To Show that f is bicontinuous, Theorem 2.9 is used. It is not hard to see that the unit sphere S of E3 has the long legged local issoceles property with x(p) =‘/31 Theorem 2.9 then shows that f is locally bicontinuous and, in particular, PM) #0 for d<\/3. We now Show that f is one-to-one. It is here, and only here, that we have not settled the theorem for the case that Figure 4.6 129 E and M have infinite dimension. Let d1.\/3'g.d1 g.2 be given. Then there would be points x,y,z of S as in Figure 4.6, where d2 is some distance less than \/3 . If p(dl) = 0, it follows that p(dz) = O, contradicting p(d) so for all d<fi. We have now Shown p(d) ¥ 0 for any d, hence f is one-to-one. To Show f is onto, an argument such as the one in Corollary 4.16 can be used. Hence, f is bicontinuous. Next we show that f(-p) = -f (p) for any peS. Let p be an arbitrary point of S and let A = [x ES [p(d(p,x)) = 2} and B = [yeU] “f(p) -y|] = 2]. Then f(A) = B. so the sets A and B, as well as their complements, are homeomorphic. Figure 4.7 The set B can be shown to be a convex subset in U. The set A must consist of a family of parallel "lines of latitude" with p as the south pole). As A is homeomorphic to B, A mustbea spherical cap and must contain -p. 130 If A is run: equal to {-p}, then every Spherical cap the same size as A is mapped onto a convex set in U. Clearly this leads to a contradiction. Thus A = {—p], and B = [f(-p)]. Since -f(p) EB, it follows that f(-p) = —f(p) . To Show p is one-to-one, a similar approach is taken. Let A = [x68 lp(d(p,x)) = r] and B = {yEU I d(f(p),y) = r]. As before, A and B are homeomorphic and A contains a number of lines of latitude of S. However, B can be shown to be a Simple closed curve in U (for r < 2). Thus A must consist of one line of latitude, so p is one-to-one. The proof that f is an isometry can now proceed as in Theorem 4.15. U The questions raised in this section could be aSked of order and metric transformations between the spheres of arbitrary N.L.S. We have made no progress on this type of question. While considering it, we aSked the following: If U1 and U2 are the unit Spheres of N.L.S. M1 and M2 respectively, and onto 1 other words, can f be extended to M1? This problem seems to f :U > U2 is an isometry, is M1 isometric to M2? In be quite difficult, and yet should be easier than questions about metric and order transformations between spheres. If such theorems are true’they would supercede theorems of the Mankiewicz type discussed in Chapter 3. 131 §5. Appendix to Chapter 4 In proving Theorem 4.5, the following fact was used. Let V ‘2 2 h(o) = 2‘, Bj sin hjo 3=1 s 9(a) = 4r2c2sin2 §+ Z] Agsinsmkjr sin 3) , j=1 . where sj'hj'c'Aj' and kj are all non-negative constants. If h(o) = g(o) for all a 2_O on a non-empty interval I, then for each j = l,...,s either Aj or k3. is zero. The following proof of this is due to Dr. L. Sonneborn. Assume that h(o) = g(o), for erI. Consider h and g as functions of a complex variable 2. Both are analytic funtions on the entire complex plane. By a well known theorem of complex analysis, if h(z) = g(z) for all z in any set having an accumulation point, then h(z) = g(z) for all 2. Since we know that h(o) = g(o) for every a in I, it must be the case that h(z) = g(z) for all 2. B2 '11 (e ‘2hjt _ 2 + eZhjt) 132 Let b = max[hj]. Then 1im e ‘(2b+1)th(it) = 0 t4- -(2b+1)t 22 2 it Similarly, 1im e -4r c sin 1? = 0. New t-nu -(2b+l)t S 2. 2 . it A.s n 2k.r s e .2) J 1 ( j in 1?) 3=1 . . it . . it -(2b4-1)t s 2 e12k3r Sin 1?..e-lsxjr Sin 1? 2 = e Z) A.(-4 4;. ) j=1 3 2i . -t/2 t/2 . -t/2 t/2 _ -(2b+l)t S 2 ez'br‘e 'e )-2+e"2"3r(e “3 ) - e Z A- ( ) j=1 J ‘4 Taking the limit as t-4a, we get -a. unless either Aj or kj is zero for each j,j = 1,...,s,. From the above we know this limit must be 0, so .Aj or kj is zero for each j. a CHAPTER 5 FURTHER RESULTS IN EUCLIDEAN SPACE In this chapter we consider metric transformations 1, n 2.2. Since we between hypersurfaces immersed in En+ make use of results from differential geometry, the hyper- surfaces are required to be smooth, and the metric trans- formations to be diffeomorphisms. It is not our intention here to carry out a complete investigation of diffeomorphic metric transformations, but rather to show some applications of the theory and techniques outlined in chapters one through four. We suspect that there is much room for generalizing the results of this chapter. Before proceeding, a word of caution. Normally, a differential geometer calls a function which preserves arc length an isometry. In our work we consider, for the most part, hypersurfaces in En+l, and continue to mean by an isometry a function which preserves the Euclidean metric. A map which preserves arc length is called eguilong. The links connecting preceeding chapters to differential onto> '5' be a geometry are Theorems 2.14 and2.20. Let f : S (diffeomorphic metric transformation with spread 1. Theorem 2.14 ashows that,for any rectifiable arc y in S, the length of 133 134 y equals the length of f(y). That is, f is equilong. Thus, the first fundamental forms of s and s are the same. (See §l for definitions, symbols and references). Theorem 2.20 shows that, for a differentiable curve y in S and a point pesy, the difference of the squares of the curvatures of y at p and of f(y) at f(p) is constant. The constant is independent of both Y and p. This enables us to relate the second fundamental forms of S and 'S and, in the case of hypersurfaces in En+1, to show that they must be the same, up to Sign. See Theorem 5.6. Because f, a metric transformation with spread l, preserves the first fundamental form (ie.arc length) a large body of knowledge is immediately applicable. This body of knowledge, known as "rigidity theory", investigates diffeomorphisms between submanifolds of a manifold. The diffeomorphisms are assumed to preserve the first fundamental form. The question aSked is whether or not such a diffeo- morphism can be extended to a diffeomorphism from the manifold to itself, with the extension also preserving the first fundamental form. One of the most famous rigidity theorems is the following. Theorem 5.1: (Cohn-Vossen) If S c E3 is a compact surface which is the boundary of a convex set, then every equilong diffeomorphism of S into E3 is an isometry. 135 Proof: See [36] 5, p 280 Theorem 12. It follows immediately from Theorem 5.1 that a diffeomorphic metric transformation, defined on the boundary of a compact convex set in E3, into Es, is a similarity. Another famous rigidity theorem, used later in this chapter, is the following. Theorem 5.2: Let s and s be differentiable hyper- surfaces immersed in En+1, and let f :S 9259) S. be an equilong diffeomorphism. .Assume also that the rank of the differential is at least 3 for all points of S. Then f is an isometry. Proof: See [36] 5, p 244 Theorem 1. Theorem 5.10, the main result of this chapten eliminates the assumption of Theorem 5.2 that the rank of the differ- ential be at least 3. 1 (such The rigidity theorems for hypersurfaces in En+ as 5.1 and 5.2) in themselves seem to support the principle of "metric determinacy". That is, they seem to confirm that an order transformation is "usually" an isometry. (See Chapter l, 53). Theorem 5.10, and the results of Chapter 4, may make the reader feel very comfortable with the principle of metric determinacy, for subsets of En+1. However, questions about metric transformations between submanifolds of dimension less than n, in En+1, have not been addressed. For example‘ we have not determined whether or not a metric 136 transformation of an n-1 dimensional manifold immersed in En+1 is necessarily an isometry. The case of a submanifold of dimension one is partic- ularly interesting. In Chapter 4, all the metric trans- formations of the real line into En are classified. It is easily seen that if a continuous metric transformation of the real line has an inverse which is also a metric transformation, then that metric transformation is necessarily an order transformation. If f1 is an order transformation of JR into En, f2 a metric transformation of JR into Em. Then f2 afil is a metric transformation from a curve in En into Em. One may aSk whether or not this is the only type of metric transformation from a curve in En into Em. The answer to this is by no means clear. The remainder of this chapter is divided into 2 sections. Section 1 contains a list of symbols, some definitions, and a few elementary lemmas. Section 2 contains the main results of this chapter, in particular Theorem 5.10. §l. Notation, Definitions, and Elementary Lemmas. Throughout this chapter S and S. (note that S. is not, as in Chapter 4, the closure of S, but a new symbol) are n-dimensional differentiable hypersurfaces immersed in n+1 E (n 2_2). The function f :3 onto >'§ is a diffeomorphic metric transformation and, as usual, p is the scale 137 function associated with f. For any U c S, 5'- f(U) and in particular, for x ES, 32 5 f(x) . If v is any vector tangent to S at p, then 3' is the corresponding vector tangent to S. at 5) under the map between the tangent spaces induced by f. We have tried to keep the notation and terminology "standard" in this chapter. For the readers convenience we now present a list of symbols used in this chapter, along with their meanings. we refer the reader to Spivak [37] for any terms he may wish defined in more detail. Let peas, and y c S be a differentiable arc with Pey. k (p) The curvature of the curve y at p, as Y defined in differential geometry. Called the "classical curvature" in Chapter 2. S The tangent space to S at p. P The symbols N(p), de, Kn(p,V), and up are only defined for S an orientable hypersurface. N(p) The unit normal at p in the given orientation of S. dN The differential of N at the point p, p called the Weingarten Map. Let u,v6S . KN(p,V) The normal curvature of S at p in the direction v. I (u,v) The first fundamental form of S at p, p evaluated at u and v. np(u,v) The second fundamental fonm of S at p. evaluated at u and v. 138 d(y) The length of the arc y. n(p,o) The normal to the curve a at p. Let vl,...,vK be vectors in En+l. ”v1” The standard norm in Euclidean space. The inner product of v1 and v2. [vl,...,vK] The subspace of En spanned by v ,...,v . 1 k 1 _ - [v1,...,vK] - [u|— O, ve[v1,...,vK]]. Note that a symbol with "bars" is the corresponding symbol in ‘S. For example rfi4535) is the normal curvature of 'S at p in the direction 51 Equilong maps were referred to earlier in this chapter. Their formal definition now follows. Definition: A diffeomorphism f : s °nt°> s is said to be guilong if and only if Ip a- Ip for all p e S. A standard result of differential goemetry shows that an equilong function preserves arc length. As noted earlier, differential geometers use the term isometry, rather than equilong. In this chapter, f is an isometgy will continue to mean d(x,y) = d(f(x),f(y)) for all x,yes. Throughout this thesis, we have considered a metric transformation, or order transformation, f. We have tried to assume as little as possible about f, preferring to make hypotheses about the domain and range, rather than the 139 mapping itself. For example, although we often assume f to be continuous and to have finite spread, Theorem 2.9 and 2.16 show that this is true if the domain and range contain rectifiable arcs. In this chapter, we are not so fortunate. For Theorem 5.10, our main result, we aSk that f have spread l, and be a diffeomorphism. The assumption of Spread 1, rather than finite spread, allows us to prove f is an isometry, rather than a similarity, and is included for convenience. Unfortunately, we are unable to Show anything about the differentiablility of a metric transformation between two differentiable hypersurfaces. However, Theorem 5.3 shows that a metric transformation from a differentiable Euclidean hypersurface is necessarily continuous and locally bicontinuous. onto Theorem 5.3: If f :2, > i c: )3r1 is a metric transformation, where Z] is a connected regular hyper- surface, then f is continuous and there is a x > 0 such that f‘B(t,)[/2) OZ: is bicontinuous for any t 62). Since none of our future work depends on this, we only outline the proof. \ 140 Proof: We need the following standard facts about a hypersurface Z) regular at a point p. 1. Z) has a unique tangent hyperplane at p and a unique normal line pn at p. For any sequence of points [pi] on Z) approaching p the measures of angles [< n p pi] approach 90°. 2. There exists a cylindrical neighborhood U of p, with axis pn, such that U‘nZ} is a topological (n -1)-ball D and o-D consists of exactly two components. Based on these observations we can conclude that there is a cylindrical neighborhood, U, of p, axis pn, radius x such that for any x 6D = Zno the measure of angle < n p x is between 89° and 91°. This means that no point of D is in the double napped cone with vertex p, axis pn and vertex angle 89°. Now for x 6D, consider the plane n px and note that there are two points in this plane, q and q* such that pq = qx = pq* = q*x = x and further that q and q* are in o with q in one component of O'-D and q* in the other. n Finally the locus of points y in B such that py = yx = x is an (n-2) sphere which is a subset of o 141 This connected set must intersect D and thus there is a point m of Z) such that pm = mx = x. This implies that Z] has the 111 property at p. If there are two points p and q of :3 such that f(p) ¥ f(q) then there is a rectifiable arc joint p to q. Hence by Theorem 2.9 le(t,x/2) lS bicontinuous for any t 62. [:1 onto> S. be a metric transformation, and 1 Let f :S lets S,S' be smooth hypersurfaces in En+ , n 2_2. Combining Theorems 5.3, 2.9 and 2.16, it is seen that f is necessarily continuous, locally bicontinuous, and has finite spread. Unfortunately, since we require f to be a diffeomorphism, this information is superfluous for our present purposes. The following two lemmas are used to prove Theorem 5.6. They are standard results of linear algebra. Notation: Let T be a linear transformation. We write Tx for T(x). n Lemma 5.4: Let T :En 4 E be a symmetric linear transformation. Let el be such that [] = max [] 1 ' ”xu=1 5 142 Then e1 is an eigenvector of T, with the corresponding eigenvalue. Proof: Since T is symmetric there is a set of orthonormal eigenvectors of T spanning En, call them [b.]s_ , with Tb. = k.b.. Let A be the matrix for T 1 1—1 1 i 1 in the basis [bi], Let x=2o.b., with 2o§=1, (i.e. Hx”=l). Then 1 i = Z)kici. From this we can see that both max IIXH=1 and min (Tx,x> are eigenvalues, and the values of x HXH=1 where these are obtained are eigenvectors. Thus it follows that if ] = max , ‘ 1 1 qu=1‘ ‘ then e1 is an eigenvector, and is the corre- sponding eigenvalue. D Lemma 5.5: We can define a complete set of eigenvectors for the symmetric linear transformation T as follows. Let e1 be such that [] = max [| . HxH=l 143 Assume e1,...,eK_l have been defined, k g_n. Let eK be defined by [] = max [[ . k k HxH=l ‘ xe[e1,...,eK_1] Proof: This follows immediately from Lemma 5.4 by noting that for k g_n. T is a symmetric = T. J. K [81,...oeK] linear transformation on [e1....,e 1‘, and that any eigenvalue or eigenvector of TK is an eigenvalue or eigenvector respectively of T. [j 62. This section contains the main result of this chapter, Theorem 5.10. As has been mentioned, Theorem 5.10 shows that a diffeomorphic metric transformation f, with spread 1, between connected hypersurfaces S and S. in En+1, is an isometry. Theorem 5.6 contains the crucial argument: that, up to Sign, f preserves the second fundamental form. If peas is such that up a 0, standard theorems of differential geometry then Show that :f is, in a neighborhood of p, an isometry. Theorem 5.9 then shows, by purely metric means, that f itself must be an isometry. Theorem 5.6: Let S be an orientable differentiable hypersurface immersed in En+l. Let f :S onto> SEEn+1 be a diffeomorphic metric transformation with spread 1. Then, 144 for any p65 and u,vesp, 355.?) = np(u.v) or 313(6)?) = -np Ku(p) (2) - kN(p.c’(0)) More generally, for u,v esp, np(u.V) = - To prove Theorem.5.6, it suffices to Show that dNS a i dNb. 145 First we show $5,?) = :tflffipfl) +c for each p68, vessp ,“VH = 1. In the following, d(s) is a curve with d(O) = p and a’(O) = v. New [kN(p.v)l = [ka(p)| l| As there are curves d(s), (for example, the normal section), with |] = l, we can write (3) lkN(p. V)]= am(in) Ina (p)| s From (1) it follows that (4) [KN(p,v)|a_1_n(isr; [k-(p) [= min ./K§ (p) + c . d(s) Clearly the minima (3) and (4) occur for the same curve d(s). Hence, [$(EOV)] =\/‘§(P0V) +C Next, apply Lemmas 5.4 and 5.5 to the symmetric trans- formation dN p : IRn-aRn and define the eigenvector el of de by [kN(p.e1)] = l] = Hxn=1 ‘l s 1144 “(NW ...] Let [kN(p,el)] = K1. Thus el is an eigenvector of dNP and k1 the corresponding eigenvalue. Since f preserves the first fundamental form, ”x” = H§fl and 146 -—--— -—‘— 2 max || = max RBI-(pm) I = max \/KN(p,X) + c ”xu=1 P Hxn=1 qu=1 This maximum occurs for the same value of x as the prece- eding maximum, that is for x = e1. Hence “gfixl «figs» = 11=/x§ + c X = Thus, using Lemma 5.4, ‘31 is an eigenvector, corresponding (defn) to the eigenvalue x1 == (dEEEi,Ei>. From the above, 2 ,/2 E1 = i/K'Nm'el) +c = i x1+c Using Lemma 5.5, and an argument as above, choose e2,...,en eigenvectors of de, corresponding to the eigenvalues xi = KN(p,ei) while e2....,en are eigen- vectors of dfia corresponding to the eigenvalues ._ _ 2 _ _ Ki - :./Ki4-c . In the bases el,...,en and e1,...,en we have K1 0 x1 0 dN = dfi— = P O Kn O Kn Assume that de has ranx at least 3. Since de is a continuous function in p, and a differentiable hyper- surface is locally connected, there is a neighborhood Us:S of p such that qu has ranx at least 3 for all quU. Remembering that f is equilong, and applying Theorem 5.2, "x 147 we see that f ‘U is an isometry (i.e. an Euclidean motion) and hence up a £fi3° Now assume that de has ranK at most 2 and that K3 = ... = Kn = 0. By the Theorema Egregium ([35] 4, p 98, Corollary 23), 1K1K2.. .K n|=]K1K .K nl=fi +c./K2 +c Ki-l-C Since c 2_O, it is immediately clear that c = 0. Hence ifil O __ :HCZ db?— = o p C On the other hand, the set [Ki Kj [i < j} .is identical, including multiplicities, to the set {Ki Kj ‘i < j}. ([36] 4, p 97, Proposition 22). From this it follows that Kle = KlK2 and hence either K1 0 -K1 0 K ._ -K —- 2 0 or dN—= 2 o P P 0 O O O The proof of Theorem 5.6 is now complete. D Corollary 5.7: If, in addition to the hypotheses of Theorem 5.6, S is connected and up does not vanish identically for any peas, then f is an isometry. 148 Proof: As H is continuous in p, up i O for any p, and S is connected, we must have either “p ‘11:; for all p 68 or Hp 5 -EE- for all p 55. Remembering that f is equilong, the corollary now follows from [36]. 4, p 23, Theorem 21 . [:1 Lemma 5.8: Let V be a connected subset of a metric space M. Let peM. q1,q2 6V and assume d(p,ql) 0. Our first objective is to show that d(p,q) = d(f(p),f(q)) for all p as, q 6U. Let qO be a fixed but arbitrary point of U. On the line normal to U at q0 there are 1 . n+ _ _ only two points xl.x2eS’. Assume d(p,qo) 2.do- Then by Lemma 5.8 there is a I ~ g - p068 With d(po,qo) dO' The line through p0 and qo is not normal to the hypersurface U at qo. Thus there are points ql and q2 of U with d(po,ql) < do==d(po,q0)<( d(po.q2). Consider B = {x ‘d(po.x) < do}¢1U. Since nq ¢ 0 for qegB, B does not lie in an n-flat of En+1, hence 1 is a metric basis of En+ (see Chapter 4, g1). By the definitions of B and p0, is an isometry. As f 'BtJ{pO} f 'U is also an isometry and Bs:U, the above remarK shows that f ‘ULJBLJ[PO} = f ‘ULJ{PO} IS an isometry. 150 Let d be such that d(po,ql) < d < d(po,q2). By Lemma 5.8. there is a qu with d(p0,q) = d. As f ‘ULJ{PO} is an isometry then P(d) = D(d(P0:q)) = d(f(PO).f(q)) = d(po.q) = d It now follows that p(d) = d for all d < d(po,q2), showing that dO 2.d(po,q2). This contradicts the choice of q2, hence the assumption that d(p,qo) 2_d0 is false. However if d(p,qo) < dO .it follows from the definition of :10 that d(p,qo)=d(f(p),f(qo)) for all peS’. If either x1 or x2 is in S - say xj, the continuity of f shows that d(xj,q0) = d(f(xj),f(qo)). As qO is an arbitrary point of U, it follows that p(d(p,q)) = d(f(p),f(q)) for all p68, qu, and hence f ‘UU {p} is an isometry for each p es. The above remarK now shows that f is an isometry. [3 Theorem 5.10: Let S be a connected differentiable 1 1 onto) S c; En+ a diffeomorphic hypersurface in En+ , f :S metric transformation with spread 1. Then f is an isometry. Proof: Any differentiable hypersurface is locally orientable, hence locally one may consider up. If up a O for all pES, then by Theorem 5.6, fisvs up 5 O, for all peas. Then S- and S. both lie in n-flats of En+1. Hence Theorem 5.10 follows from either Theorem 4.7 or Corollary 3.l9(a). 151 Otherwise there is a qO es such that (in a local orientation) an i 0. As H is continuous in qo, and a differentiable hypersurface is both locally arcwise connected and locally orientable, there is an open arcwise connected subset U of S, containing go, for which we can assign an orientation N(p) such that nq )4 O for all q 6U. Noticing that any connected manifold S of dimension at least 2 is such that S\[x1,x2} is connected, for any x1 and x2, we now apply Lemma 5.9 to complete the proof of Theorem 5.10. I] 1 1 Corollagy 5.11: Let f : S 4 ScEm' be a diffeomorphic metric transformation, 8' a differentiable hypersurface in En+1. Then f is a similarity. We thinK that Theorem 5.10 has many possibilities for generalizations, the most promising may be to submanifolds, rather than hypersurfaces, immersed in En+l. Two of the principal theorems from differential geometry that we use in the proofs of Theorem 5.6 and Corollary 5.7 ([36] 4, p 97, Proposition 22 and [36] 4, p 93, Theorem 21) remain valid for both hyperbolic and spherical space,and one might suspect that Theorem 5.10 could be generalized to hypersurfaces immersed in manifolds of constant curvature. However Example 7, Chapter 1, shows that this is not the case . CHAPTER 6 EXISTENCE QUESTIONS This chapter is an attempt to summarize and extend some of the worK that has been done on the "Existence Question". The "Existence Question" was discussed in Chapter 1. Essentially, it is this: Given a distance space (N,d) and a class a. of distance spaces, is (N,d) order embeddable onto some member of 6? Frequently, c. is taKen to be all subsets of some particular distance space - say En. All of our results are of this type. The chapter is divided into two sections. Section 1 deals exclusively with finite sets, while Section 2 deals primarily with infinite sets. In both sections, most of the results are of a "negative" type. That is, they show that a certain space is not order embeddable into a certain class cu §l. Of most interest to the MJD.S. Theorists are finite sets of points, with an ordering of S xS. That is, they begin with a set S, a totally ordered set C and a mapping e :S;xS.4C. Then (S,C,e) is called a C-metrized space. Because all that is of interest is the ordering 152 153 induced on S xS by C and e, and as e(S x8) is a finite subset of C, C may always be taKen to be a subset of the positive reals. In this case, (S,C,e) is a distance space and is denoted by (S,e). One type of Multidimensional Scaling order embeds (S,e) into a metric space (M,d). One way of doing this is to find a metric d on S such that the function f taKing (S,e) into (S,dL given by f(x) = x’ is an order transformation. This is easy to do. If d is e(x,y)J-K x # y defined by d(x,y)== and K is chosen 0 x = y suitably large, then (S,d) will be a metric space. For example, if K 2_ max [e(x,y)L then for any distinct x,yeS x,y.zes. d(x,y) +d(y,z) = e(x,y) +K+e(y,z) +K 2 2K 2 e(x,z) +K = d(x,z) This shows d is a metric, but the metric space (M,d) is not very useful without Knowing something more about K. There has been some worK done on the so called "additive constant" problem, which attempts to find "apprOpriate" values for K (see [271,[38]). Deciding on the "appropriate" value of K requires more detailed information. One possibility often considered is the value that minimizes K. In general the M.D.S. user would liKe to embed (S,e) into a metric space he Knows - such as a Euclidean space or a normed linear space. Naturally, he would prefer a Euclidean space. We now 100K at a few results in this area. 154 If S contains n points, (S,e) can be order embedded into En‘l l as follows. Consider an equilateral simplex in En- Any edge can be either shortened or lengthened slightly by a rotation about the opposite (n-2)-face, without changing the lengths of any of the other edges. Clearly, the lengths of these sides could be ‘ arranged to correspond to any ordering of S xS. If the length of the longest side is attained for only one pair of points, it can be lengthened until the simplex lies in an n-l l (n -2)- flat of E . These remarKs apply also to Hn- and Sn-l. In [19] Holman characterizes all the distance spaces (S,e) of n points which can be order embedded into En-Z. He shows that if for some x,y,z in S the inequality (l) e(x,z) g_max[e(x,y),e(y,z)] is not satisfied then (S,e) is order embeddable into En-2. The inequality e(x,z) g_max[e(x,y),e(y,z)] is often referred to as the ultra-metric inequality. It is easily seen that the ultra-metric inequality implies the triangle inequality, and that it is satisfied if and only if all the triangles in the space are "long legged isosceles"; that is if and only if all triangles are isosceles and the length of the two equal sides is greater than, or equal to, the length of the third side. Thus the only metric n-tuples not order embeddable are the ultrametric n—tuples. 155 The presence of equal distances is clearly a complicating factor in the geometric embedding problem, as well as in the various scaling interpretations. Thus we consider only scalene configurations for the time being. A distance space (S,e) is said to be scalene if and only if e(x,y) 94 e(z,w) for any x,y,z,weS with [x,y] 54 {z,w], x # y. Holman's Theorem implies that any scalene distance space (S,e) of n points can be order embedded into En-Z. The problem of order embedding an arbitrary scalene config- uration of n—points into lower dimensional spaces now arises. In general, we have not been able to maKe much headway against this question. We now present some results, first considered by Kelly and Erdos (unpublished), which answer the question for n = 4,5, and 6, and give a bound for the minimum dimension needed for larger n. The case n = 4 is easy, so it will not be considered here. For n = 5 and 6 an example is presented which cannot be order embedded into E2 or E3 respectively. Thus an arbitrary 5 or 6 point scalene configuration need not be order embeddable into E2 or B3 respectively. Unfortunately, obvious generalizations of these examples to n = 7 are order embeddable into E4. The result for n = 5 (Theorem 6.1) shows the stronger fact that there exists a scalene distance space of 5 points not order embeddable into any two dimensional normed linear space. 156 Theorem 6.1: Let (S,e) be any distance space, s = {p1,p2,p3,ql,q2}, such that (2) e(Pion) > e(qllqz) > e(pm’qn)’ i 5‘ j Then (S,e) cannot be order embedded into any two dimensional normed linear space. Proof: Let f :S-+E2 be an order embedding of S. For simplicity's saKe, call f(x) simply x. Figure 6.1 Consider the above diagram. We begin by assuming that one of the points q1,q2 is in one of the closed regions 157 labeled 1,2, or 3. Say q1 lies in 1. We have q1P3 + (111? Z P3131 + Ply szi'Ply 2.P1P2 Adding these, and simplifying, we obtain q'lp3 + qu + pzy 2 133131 + 19192 or However, this contradicts (2), since f is to be an order embedding. Assume neither q1 nor q2 lies in any of 1,2, or Figure 6.2 158 Then there is a pair of points pi,pj, such that the line qlq2 does not intersect [pi,pj] and the line pipj does not intersect [q1,q2]. The pOints pi,pj,q2,q1 form a convex quadrilateral with [q2,ql] and [pi'Pj] being opposite sides. Say the quadrilateral is p1p3q2ql. By Lemma 3 .80 d(p1.p3) +d(q2.q1) g d(p1.q2) +d(p3.q1) Hence, one of the terms on the left is less than or equal to one of the terms on the right, contradicting (2). A similar contradiction is obtained in all cases. This completes the proof. [3 We have been able to generalize this to 6 points only for order embeddings into E3. Theorem 6.2: Let (S,e), S = [p1,p2,p3,ql,q2,q3), be any distance space such that (3) e(Pi'Pj) > e(qx.q‘) > e(pmoqn). i e j. K # 1 Then (S,e) cannot be order embedded into E3. Proof: As in Theorem 6.1, 1et f :S.-.E3 be any order embedding of S, and call f(x) simply x. Let V be any plane containing pl.p2. and p3. 159 .r /p3 p2\ Figure 6.3 At least two of q1,q2,q3 lie on the same side of w, say ql and q2. Let qi,q£ be the perpendicular projections of g1 and q2 onto w. The situation in v is very similar to that in the previous theorem, using qi,qé instead of q1 and q2. ‘The only difference is that d(q£,qé) g d(ql,q2), so d(qi,qé may be less than d(pi,q5) for some i,j. However, as in Theorem 6.1 it can be shown that I I I I d(pm.pn) + d(q2.ql) g d(pm.q2) + d(pn.ql) for some m,n = 1,2,3, m # n. By (3), d(pm,pn) is the largest of these distances (for d(pm,qé) g_d(pm,q2) and 160 I I I d(pn,q1) g d(pn.q1)). so d(q2,q1) must be the smallest. I I I I I I It follows that d(pm,q2) 2,d(q2.q1) and d(pan1)2,d(q2.ql Assume d(q2,qé) 2_d(q1,qi), (otherwise change the roles of q1 and q2, as well as pn and pm in the following ). Then 2 _ , 2 , 2 d(pm.q2) - d(pm.q2) +d(q2.q2) I I 2 I I 2 2 d(q2.q1) + (d(q2.q2) -d(q1.q1)) _ 2 - d(q2.q1) This contradicts (3) so the proof is complete. D As has been said, there seems to be no obvious generalization of Theorems 6.1 and 6.2, even to n = 4. The next result shows that if every distance space of n points is order embeddable into some N.L.S. of dimension. 1° ((32-2) gm. This shows m, then it is necessary that that,as n tends to a, so does the minimum dimension such that every distance space of n points is order embeddable into a N.L.S. of that dimension. Theorem 6.3: In order for every scalene distance space of n points to be order embeddable into some N.L.S. M of dimension m, it is necessary that 1 ($1-2) £111 and sufficient that m g n -2. Proof: That it is sufficient that m g_n-2 follows from Holman's Theorem. 161 Consider any distance space (S,e) consisting of the n points {p1,p2.q1,...,qn_2} such that (4) e(p1.p2) > e(qi.qj) > e(qx.p‘). i # j . As in Theorems 6.1 and 6.2, let f :S.+M be an order isomorphism, and denote f(x) by x. Let r = d(p1.p2). Because the triangle inequality is satisfied, d(qi,p1)4-d(qi.p2) 2_d(p1.p2) = r for all i From this, and (4) it follows that r . . d(qi,qj) > 2 for all i and 3 Again because of (4) . qi E B(p1,r) for all 1. Thus B(qi,r/4)s:B(p1,5r/4) for all i. Also, because d(quqj) > r/2 it follows that B(qi,r/4)r13(qj,r/4) = ¢ for all i,j. Let Vt be the volume of a ball of radius t. Then (n-2)Vr/4 S-VSr/4' Now Vt = th1 (see [5 ], page 158), so (n--2)(r/4)m g_(5r/4)m. Thus, it follows that lo (n-2) mZ—fa'r- D Clearly, the above theorem could be improved. However we have not been able to obtain any lower bound on m which is not a logarithmic function of n. At least for Euclidean space, it should be possible to do better than this. 162 §2. In Section 1 we presented a few results concerning the existence question for finite sets. We now turn to infinite sets. In [24] Lew gives "Some Counterexamples in M.D.S.". He shows the following: (1) The spaces 1?» and I: have no order embedding into En for m.2_2. However for each m, IT can be order embedded into fl, Hilbert space. (2) The space C of all real sequences with limit 0 zero, and norm defined by ”(xn)” = sup [xnl has no non- ' n trivial metric transformation into Hilbert space. Lew calls these counterexamples because they show that, in applying M.D.S" it is necessary to consider spaces other than En and y; The proofs of (l) and (2), as presented by Lew, are quite difficult, relying heavily on the worK of Schoenberg, von-Neumann, and Einhorn ([12], [29], [32]) on positive definite forms. We now present an elementary proof and generalization of the first part of (1). Theorem 6.4: A N.L.S. which has a segment on its unit sphere has no metric transform in En, Sn, or Hn. Proof: Let M be a normed linear space with a segment on its unit sphere. Consider a plane v that contains the origin, and a segment of the unit sphere with endpoints p1 and ql. It is sufficient to show that this plane, with the induced norm, has no metric transform into En,Sn, or Hn. Let qO be the Po Figure 6.4 reflection of p1 through the origin, and p0 be the reflection of q1 through the origin. Then the segment joining pO to qO also lies on the unit sphere. Let I and ‘P be the lines through qO and q1, and p0 q and p1 respectively. For i > 1, let qi be the point on ‘q which is a distance 2 from q. but is not i-l' equal to qi_2. Similarly define [pi]:=0 on ‘p' It is easily seen that (3) Hqi ‘qu = Hqi’PjH = ”Pi "PjH = ”Pi ‘qu for all i 7f 3' 164 Let M be any of En,Sn, or Hn and let f :v-aMl 1 be a metric transformation. For a,b 6M1 1et B(a,b) = [x‘d(x,a) = d(x,b), x6141}. Now B(a,b) is an (n-l)-flat of M1. (See Busemann [5] p. 309). Thus, if a ’,b’ 6 B (a,b), then [xeB(a,b) ld(x,a’) =d(x,b’)]=B(a’,b’) n B(a,b) i-l is an (n-2)-flat of M1' Continuing, if ai'bi 6 K91 B(aK,bK) , m then .r‘ B(ai'bi) is an (n-m)-f1at of M1. -\ i=1 Now because of (3) and the fact that f is a metric transformation, d(f(qi).f(qj)) = d(f(qi).f(pj)) = d(f(pi).f(pj)) = d(f(pi).f(qj)L Thus f(qi) and f(pi) are in B(f(qj),f(pj)) for i # j. m. Therefore f(qi),f(p3) 6 r) B(f(qj),f(pj)) for i > m. i=1 m Hence r1 B(f(qj),f(pj)) contains an infinite number i=1 n of points for all m. However, r] B(f(qj),f(pj)) is i=1 zero-dimensional, hence contains 1 or 2 points (1 if M1 is En or Hn, 2 if M1 is Sn). This is a contradiction. and the theorem is proved. [3 Corollary 6.5: If a N.L.S. has a segment on its unit sphere, then no open subset of that space has a metric transform in En,Sn or Hn. The proof of this is essentially the same as the above. 165 The space 12 seems to lend itself nicely to problems involving order transformations. Theorem 6.10 show that many N.L.S., in particular En, cannot be order embedded into 1:. The proof of these theorems is based on the characterization of "flat spots" of the unit sphere of a N.L.S.,given in Lemma 6.8. Before proceeding to Lemma 6.8 several definitions and preliminary lemmas are needed. In these definitions, and throughout the rest of this chapter, convex always, means algebraically convex. Definitions: Let K be a closed algebraically convex set in a n;L.S., M. A hyperplane H in M is said to support K if K lies in one of the closed half spaces determined by H. A subset S of K is said to be a supporting set of K if S = Ker for some supporting hyperplane H of K. Clearly a supporting set of K is convex. An extreme point of K is a supporting set of one point. Supporting sets of more than one point are called §§g§§_of K. A face of K which is prOperly contained in no other face of K is called a facet of K. Definition: An affine subspace of M is a translate of a linear subspace of M. For any set A in a N.L.S. M, aff A is defined by aff A = [x [x II D15 7 P 9.! P [as y II |'-‘ :3 /\ 8 (U H. m 3’ 166 A point pegA is said to be a relative interior 2932; of A if p 6V :A, and V is open in the relative topology of aff A. If K is algebraically convex it is not hard to show that the relative interior of K is non-empty (see [14], page 9). For the following worK (Lemmas 6.6, 6.7, 6.8, 6.9) let B be the closed unit ball, and U the unit sphere of a N.L.S. M. The proofs of Lemmas 6.6 and 6.7 are omitted. Lemma 6.6: If K is a convex subset of U, then there is a supporting hyperplane H of B with K<;(aff K) nBcHnBcU Proof: See Day [8] p. 43. In other words, K lies in a face of B. Lemma 6.7: If p is a relative interior point of a facet F of B, then p lies in no other facet of B. Notationz: For p EU, define E(p) to be [x | x 6U, "p-x” = 2]. Let -p be the point of U diametrically opposite p. Certainly, -p<5E(p), hence E(p) is not empty. As mentioned earlier, Lemma 6.8, which characterizes a facet of B, is the crucial result to obtain Theorem 6.10. 167 Lemma 6.8: If F is a facet of B, then F = E(p) where -p is any relative interior point of F. Proof: Assume erF. Then, since F is convex, [‘PIX] CF - ...p y X 0 0y parallel to px Figure 6.5 In Figure 6.5, let 0y be parallel to px, with Y€E[-PIX]- Then Hy” = 1, so it follows that Hp-—x” = 2, and hence x€E(p). Thus, F:E(p). Assume x 63(1)). x (F. We will show that F could not be a facet of B, giving a contradiction. Consider Figure 6.5, where x eE(p) , and Oy is parallel to px. Then ”y“ = 1, so it follows from the convexity of B that [-p,x] cU. (Recall that x 6U, so M = 1)- 168 Let y = xx+ (l-x)q, qEF, O_<._)‘ gl I r \\\\s 1_X Figure 6.6 Consider Figure 6.6, which is the situation in the plane containing x,-p and q. As -p is in the relative interior of F, and q 6F, then r 6F can be chosen with -p,q and r collinear, and q and r on opposite sides of -p. Let [r,y]{j[-p,x] = s. Then r,s,q, and x all are in U, so it follows that yeEU. Thus the convex set (def) K = [)‘x+(l-)‘)q[qu, ogxgl];U Lemma 6.6 shows that K lies in some face of B. However F 5£K :B, contradicting the fact that F is a facet of U. Thus, E(p) c F and the proof is complete. I] Let f be an order transformation with scale function p, from U (the unit sphere of a N.L.S.) into 5:. Let K be the smallest closed ball of 1: containing f(U), 169 and let s be the corresponding sphere (that is, S is the boundary of K). Then, clearly, two opposite facets of K touch f(U). So K and f(U) have the same diameter, which is p(2). Note, we use the fact that f is an order, transformation to determine that p(2) is the maximum transformed distance. For peU, both f(p) and f(-p) lie in K and ”f(p) -f(-p)n = p(np- (-p)]|) = 9(2) Hence, f(p) and f(-p) lie in opposite facets of K. Thus, f(U)s:S. Lemma 6.9: Let f be an order transformation from U into 3:. If’ p1,...,pm are arbitrary points of U and m m m 2.2n4-1, then for some j, [J E(pi) = L) E(pi). i=1 i=1 m Proof: Let Fi’ i = l,...,r be the facets of S m containing points of f(lJ E(pi)). Let -Fi be the facet i=1 of S opposite Fi‘ Note that [f(pi); i=1,...,m]:U(-Fi) and for each K there is some members of [f(pi)] in -FK. As m.2_2n4-1 > r, for some j it must be that there is some member of {f(pi), i = l,...,m, i f j} in --FK for each K. Let f(x)eFi. Let f(px)€ {f(pi). i=1.....m. i?‘ j} be in -Fi. Then Hf(x)-—f(pK)H = p(2), hence Hx-—pKH = 2, hence x<§E(pK). The lemma is now complete. [3. 170 Theorem 6.10: Let B be the unit ball and U be the unit sphere of a N.L.S. M. If B has more than 2n facets, then U is not order embeddable into 5:. Proof: Assume B has more than 2n facets. Let -p., i = l,...,2n4-l be relative interior points of 1 2n+l distinct facets Fi' Then U E(pi) :U. By Lemma 6.9, 2n+l 2n+1 1:1 (J E(p.) = [J E(pi) for some j. However Lemma 6.7 i=1 1 i= 1an shows that -p:j (E(pi) = Fi for any i 7’ j, giving a contradiction. [3. It should be noted that a facet F may be an extreme point p. In this case aff F = aff[p] = [p], so p is a relative interior point, and Theorem 6.10 still holds. If B is strictly convex then every point of U is a facet, hence U cannot be order embedded into 1:. It is annoying that Theorem 6.10 depends on the order embedding being into 1:. These proofs do not hold up ix: other spaces, even a space such as I? which often behaves in a very similar fashion. However, Professor L.M. Kelly claims that En is not order embeddable into I?, for any n,m. CONCLUSION This thesis represents the first effort that we know' of to bring together and to expand on the geometric embedding problems underlying the data analysis technique known as multidimensional scaling. Organization of the material has been difficult and the results may seem discursive and confused. It may be well, then, to summarize and reflect on what has and has not been achieved. The principal concern has been with the study of two seemingly very general classes of transformations from one distance space to another, namely metric transformations which merely preserve equality of distances and ggggg transformations which preserve the order of the distances. It is the latter which are of prime interest to the MDS theorists. In fact it is their fervent hOpe that in many "highly structured" spaces the only order transformations are "essentially" similarities. The Beals-Krantz result [2], which we reprove in Chapter 2, supports this idea in the class of convex metric spaces. As pleasing as this result is it fails to answer such simple and relevant questions as: "does the order of distances determine a sphere in euclidean 3-space"? or 171 172 "what do order transforms of the real line look like in En"? The first of these questions is answered in Chapter 5 and the second in Chapter 2. We will now attempt to highlight what we consider the major results and achievements of the thesis making clear in the process which are more or less original. Among the more important essentially original contributions are the following: 1. The arc length, arc curvature and various continuity theorems in Chapter 2. 2. Theorem 3.16 generalizing the Mankiewicz theorem from isometric transformations to metric transformations. 3. Theorem 5.10 to the effect that two order . . . n . . isomorphic hypersurfaces in E are Similar. 4. Proof (Theorem 4.15) that any Minkowski Space whose unit sphere is order isomorphic to the sphere in En must be congruent to En. 5. There are k-point scalene metric spaces which are not order embeddable into Mp (where k is a function of n, Theorem 6.3). 6. Examples of non-similar order isomorphic hypersurfaces in hyperbolic n-space, showing that Theorem 5.10 and Theorem 4.5 of Schoenberg cannot be directly extended even to spaces of constant Gauss curvature. 173 7. The impossibility of order embedding a non-strictly convex NLS into En, Hn, or Sn (Theorem 6.4). 8. The impossibility of order embedding a strictly convex NLS into 1:. (Theorem 6.10). Much of our effort has gone into summarizing, adapting, refining and extending the work of others, notably: l. The Schoenberg-Von Neumann characterization of screw curves in En, the proof of which we have made more accessible in that setting. we have also extended this characterization to screw segments and shown that the situation in hyperbolic n-space is much more complicated. 2. The Schoenberg proof that a metric transformation from Em into En is necessarily a similarity and our extension of this result to metric transformations defined on sets in Em with a non-null interior. 3. Our simplification and extension of the work of Vogt [37] in Chapter 3, of Lew [24] in Chapter 6, and Holman [19] in Chapter 6. A number of obvious questions remain unanswered. Some of the more appealing are the following: 174 If the spheres in two normed linear Spaces are order isomorphic must the Spaces be congruent? In this connection it seems to be even unknown whether two such spaces are congruent if their unit Spheres are congruent. Is there an order isomorph of E2 in any normed linear space other than En? . n . Are there two arcs in E which are order isomorphic which are not of constant curvature? Such arcs exist in Hn. The euclidean Sphere in n-space is characterized in the class of n-dimensional Minkowski spaces by the order of its distances. Is this true of other hypersurfaces in En? e.g. is it true of the boundaries of convex bodies? Is there a finite set in a euclidean Space characterized in the class of subsets of all euclidean Spaces by the order of its distances? What is the smallest scalene space not order embeddable in E4? The perception theorist seems to Operate on the assumption that order analyzing large numbers of finite subsets of his postulated "underlying" Space will give him a clue to a characterization of the Space itself. To what extent is this true? 175 In deference to those MDS theorists and users who have perservered thus far we should concede that what we have done here possibly has little direct bearing on their concerns. The principle of metric determinacy is certainly central in scaling theory and practice but in the context of finite spaces it seldom can be claimed that the order of distances determines a set up t0~a similaritx,Scaling theorists tend to feel that if a finite metric space is order embedded in the "prOper dimensional" euclidean Space, that embedding should be unique up to an "approximate" isometry if it is properly'scaled". Furthermore, in practice, embeddings with low "stress" are tolerated on the grounds that real data is subject to stochastic uncertainties and noise. So the order of distances of a configuration is claimed to "determine" the configuration subject to much hedging. We claim that our work could be a valuable first step in trying to make the "practical principle" more precise. In the absence of that precision some of our results can be viewed as lending some support to the principle. The literature on the applications of MDS is enormous and for the reader unfamiliar with it we can do no better than to refer to the extensive and authoritative writings of R.N. Shephard. His pOpular paper [35] in Science is particularly recommended both for its attractive survey and its extensive bibliography. BI BLI OGRAPHY 10. 11. 12. 13. 14. BIBLIOGRAPHY Bachman G., Narici L., Functional Analysis. Academic Press, 1972. Beals, R., Krantz, D., Metrics and Geodesics Induced by Order Relations. Math Zeitschr. 1967, 101, 285-298. Beals, R., Krantz, D., TverSKy, A., Foundations of Multidimensional Scaling. Psychological Review 1968, Blumenthal, L.MJ, Theory and Applications of Distance Geometry. London: Oxford University Press, 1953. Busemann, H., The Foundations of MinKOWSKian Geometry. Comm. Math. Helv., 1950, 24, 156-186. Coxeter, H.S.M., Introduction to Geometry, New YorK, Wiley. Cunningham, J.P., Shephard, R.N., Monotone Mappings of Similarities into a General Metric Space. 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