On the c-concavity with respect to the quadratic cost on a manifold

TL;DR

This paper extends the theory of c-concavity in optimal transport to Riemannian manifolds, showing that certain smooth functions are c-concave under specific curvature conditions, which aids in identifying optimal transport maps.

Contribution

It proves that in Riemannian settings, functions with a Hessian less than the metric are c-concave if they are sufficiently smooth, advancing the understanding of optimal transport on manifolds.

Findings

Hessian condition implies c-concavity on manifolds

Small C^1-norm ensures c-concavity

Provides criteria for optimality of transport maps

Abstract

Pushing a little forward an approach proposed by Villani, we are going to prove that in the Riemannian setting the condition $\nabla^2 f< g$ implies that $f$ is $c$-concave with respect to the quadratic cost as soon as it has a sufficiently small $C^1$-norm. From this, we deduce a sufficient condition for the optimality of transport maps.