A study of the dual problem of the one-dimensional L∞-optimal transport problem with applications

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.

$L^\infty$-optimal transport for a class of quasiconvex cost functions

“…The question on whether there exists a dual formulation for the L ∞ -transport, similar to the now-standard Kantorovich duality, remained open until Barron, Bocea, and Jensen stated and proved a duality theorem in 2017 [2]. The theory was further developed in the 1-dimensional case by De Pascale ad Louet in [5]. Unlike in the standard integral optimal transportation, it is not immediate how to use the L ∞ -duality to prove the existence of deterministic solutions to the Monge-Kantorovich problem.…”