Calculus of Variations and Geometric Measure Theory

L. De Pascale - J. Louet

A study of the dual problem of the one-dimensional $L^\infty$-optimal transport problem with applications

created by louet on 09 Apr 2017
modified on 04 Aug 2017


Submitted Paper

Inserted: 9 apr 2017
Last Updated: 4 aug 2017

Year: 2017


The Monge-Kantorovich problem for the infinite Wasserstein distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and Jensen. We construct a couple of Kantorovich potentials which is ''as less trivial as possible''. More precisely, we build a potential which is non constant around any point that the plan which is locally optimal moves at maximal distance. As an application, we show that the set of points which are displaced to maximal distance by a locally optimal transport plan is minimal.

Keywords: Optimal transport, duality theory, cyclical monotonicity, infinite Wasserstein distance