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

## TL;DR

This paper investigates the dual problem of one-dimensional L-infinity optimal transport, constructing specific Kantorovich potentials and analyzing the structure of points displaced maximally by optimal plans.

## Contribution

It introduces a novel construction of Kantorovich potentials in 1D L-infinity transport and examines the minimal set of points displaced maximally.

## Key findings

- Constructed Kantorovich potentials that are non constant around points of maximal displacement.
- Showed the set of points displaced maximally by locally optimal plans is minimal.
- Provided insights into the dual problem's structure in one-dimensional L-infinity optimal transport.

## Abstract

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.