The MTW condition was introduced in [9] to study the regularity theory of the optimal transportation problem, and the MTW condition was used by many researchers to study other regularity properties of the optimal transportation problem. For example, the MTW condition was used by G. Loeper, A. Figalli, Y-H. Kim, R. McCann and other researchers to show Hὅlder regularity of the potential function, and by A. Figalli and De Phillippis to show Sobolev regularity of the potential function. I present two of my results about the MTW condition in this dissertation. The first result concerns about the synthetic expressions of the MTW condition. The cost function of the optimal transportation problem need a high regularity assumption (C^4) to define the MTW condition. There are some expressions of MTW condition, however, which only need much weaker regularity assumption to define, but equivalent to the MTW condition when the cost function has enough regularity. We call these conditions synthetic MTW conditions. Although the synthetic MTW conditions are equivalent to the MTW condition under some assumption, it was not shown that if the synthetic MTW conditions are equivalent under weak regularity assumption which is not enough to define the MTW condition. I present a proof of the equivalence of the synthetic MTW conditions under C^{2,1} assumption on the costfunction in chapter 3. The other result is about the Hὅlder regularity of solutions to generated Jacobian equations. In generated Jacobian equations, we study more general structure than the optimal transportation problem. Some examples of generated Jacobian equations which is more complicated than the optimal transportation problem can be found in geometric optics problems. The Hὅlder regularity result was proved by G. Loeper in [8] in the optimal transportation problem case and this can be generalized to generated Jacobian equations. Since the structure of generated Jacobian equations has more non-linearlity than the structure of the optimal transportation problem, however, there are some difficulties to apply Loeper's idea to generated Jacobian equations. We discuss about the difficulties and suggest a way to go around the problems in chapter 4. Then I generalize Loeper's idea to more general generated Jacobian equations and show that we can have a similar local Hὅlder regularity result.
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