Constructing optimal maps for Monge's transport problem as a limit of strictly convex costs

作     者：Caffarelli, LA Feldman, M McCann, RJ 

作者机构：Univ Texas Dept Math Austin TX 78712 USA Univ Wisconsin Dept Math Madison WI 53706 USA Univ Toronto Dept Math Toronto ON M5S 3G3 Canada 

出 版 物：《JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY》 (美国数学会志)

年 卷 期：2002年第15卷第1期

页      面：1-26页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Monge-Kantorovich mass transportation resource allocation optimal map optimal coupling infinite dimensional linear programming dual problem Wasserstein distance 

摘      要：Given two densities on R^n with the same total mass, the Monge transport problem is to find a Borel map s : R^n → R^n rearranging the first distribution of mass onto the second, while minimizing the average distance transported. Here distance is measured by a norm with a uniformly smooth and convex unit ball. This paper gives a complete proof of the existence of optimal maps under the technical hypothesis that the distributions of mass be compactly supported. The maps are not generally unique. The approach developed here is new, and based on a geometrical change-of-variables technique offering considerably more flexibility than existing approaches.