Strong c-concavity and stability in optimal transport

TL;DR

This paper introduces strong c-concavity in optimal transport, providing a differential criterion for stability results applicable to complex cost functions, with implications for problems like reflector design and curvature measures.

Contribution

It defines strong c-concavity, links it to stability in optimal transport, and offers a differential criterion for verifying this property under Ma-Trudinger-Wang conditions.

Findings

Established the importance of strong c-concavity for stability.

Provided a differential criterion for strong c-concavity.

Applied results to reflector and Gaussian curvature problems.

Abstract

The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we introduce the notion of strong c-concavity, and we show that it plays an important role for proving stability results in optimal transport for general cost functions c. We then introduce a differential criterion for proving that a function is strongly c-concave, under an hypothesis on the cost introduced originally by Ma-Trudinger-Wang for establishing regularity of optimal transport maps. Finally, we provide two examples where this stability result can be applied, for cost functions taking value +$\infty$ on the sphere: the reflector problem and the Gaussian curvature measure prescription problem.