Some reverse inequality in optimal mass transportation

TL;DR

This paper establishes a new reverse inequality relating different Wasserstein distances in optimal transport, especially for costs decreasing with distance, broadening the theoretical framework for such inequalities.

Contribution

It introduces a general framework for reverse inequalities in optimal transport with decreasing pointwise costs, extending previous results for increasing costs.

Findings

Proves reverse inequalities for $ ext{W}_ ext{infty}$ and $ ext{W}_p$ Wasserstein distances.

Extends the theoretical understanding of optimal transport with decreasing cost functions.

Provides a unified framework encompassing previous specific cases.

Abstract

Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.