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

arXiv: https://arxiv.org/abs/1704.02730 · HTML: https://mlsanity.com/paper/1704.02730/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2017.

## Built on by (most cited first, 1 of 1)

- [Protecting Global Properties of Datasets with Distribution Privacy Mechanisms](https://mlsanity.com/paper/2207.08367/) (2022)

## Abstract (arXiv)

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.