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

Luigi de Pascale (1) , Jean Louet (2)

1 DMA - Dipartimento di Matematica Applicata [Firenze]
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales

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.

Mots clés: Optimal transport; Cyclical Monotonicity; Duality theory; Infinite Wasserstein distance

Domaines: Optimisation et contrôle [math.OC]

Luigi de Pascale, Jean Louet. A study of the dual problem of the one-dimensional L-infinity optimal transport problem with applications. 2017. ⟨hal-01504249v1⟩