The local geometry of maps with c-convex potentials

TL;DR

This paper introduces a new regularity condition for optimal transport maps that requires only minimal derivatives of the cost function and weak convexity assumptions, broadening applicability in measure spaces.

Contribution

It establishes a regularity criterion based on three derivatives of the cost, removing the need for strong convexity and requiring only bounded densities, thus extending previous results.

Findings

New regularity condition for optimal transport maps

Equivalent to weak Ma-Trudinger-Wang condition for $C^4$ costs

Relaxed convexity assumptions on target measure support

Abstract

We identify a condition for regularity of optimal transport maps that requires only three derivatives of the cost function, for measures given by densities that are only bounded above and below. This new condition is equivalent to the weak Ma-Trudinger-Wang condition when the cost is $C^4$. Moreover, we only require (non-strict) c-convexity of the support of the target measure, removing the hypothesis of strong c-convexity in a previous result of Figalli, Kim, and McCann, but at the added cost of assuming compact containment of the supports of both the source and target measures.