Abstract
The Monge-Kantorovich problem for the W∞ 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 in [2]. We construct a couple of Kantorovich potentials which is non trivial in the best possible way. More precisely, we build a potential which is non constant around any point that the restrictable, minimizing plan moves at maximal distance. As an application, we show that the set of points which are displaced at maximal distance by a “locally optimal” transport plan is shared by all the other optimal transport plans, and we describe the general structure of all the one-dimensional optimal transport plans.
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