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

de Pascale, Luigi; Louet, Jean (2017-08), A study of the dual problem of the one-dimensional L-infinity optimal transport problem with applications. https://basepub.dauphine.fr/handle/123456789/16901

Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01504249
Date
2017-08
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
17
Metadata
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Author(s)
de Pascale, Luigi
Dipartimento di Matematica Applicata [Firenze] [DMA]
Louet, Jean
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Centre de Recherche Cardiovasculaire de Lariboisiere
Abstract (EN)
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.
Subjects / Keywords
Duality theory; Optimal transport; Cyclical Monotonicity; Infinite Wasserstein distance