几个最优映射存在唯一性定理的统一证明

Several Results About Existence and Uniqueness of Optimal Maps in Transportation Problems:a Unified Scheme Proof

基于凸锥的性质以及测度理论,本文给出了几个最优映射存在唯一性定理的统一证明.著名的Brenier定理以及其它几个与光线反射、折射有关的费用函数所对应的最优质量运输问题解的存在唯一性定理可以视为本文主要定理的重要推论.与Brenier定理的原始证明比较而言,本文证明过程简洁明了.

Based on measure theories and convex cones, we give a unified and concise theorem which proves existence and uniqueness of optimal transport maps. Some interested results can be seen as corollaries of this unified theorem, such as the Brenier＇ s theorem and some Monge＇ s problems with cost functions coming from far field reflector problems and refraction problems.