On the cost-subdifferentials of cost-convex functions

TL;DR

This paper establishes the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions in optimal transport, under certain assumptions, and improves upon previous regularity results by relaxing geometric conditions.

Contribution

It provides a direct geometric proof of subdifferential equivalence and extends regularity theory to more general settings, including the round sphere.

Findings

Equivalence between cost-subdifferentials and ordinary subdifferentials under assumptions A0-A3W.

Connectivity of contact sets of optimal transport maps.

Relaxation of geometric assumptions on domains and targets.

Abstract

We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma, Trudinger, and Wang. The connectivity of contact sets of optimal transport maps follows as a direct corollary. Our approach is quite different from the previous result of Loeper which he obtained as the first step toward his Hoelder regularity theory of potential functions, and which was based upon approximation using the regularity theory of Ma, Trudinger, and Wang. The result in this paper improves his result, by relaxing certain geometrical assumptions on the domains and targets; it also completes his Hoelder regularity theory of potential functions on the round sphere, by making it self-contained.