A Remark on the Potentials of Optimal Transport Maps

TL;DR

This paper provides simple sufficient conditions for a smooth function to be c-convex in optimal transportation problems with costs derived from Lagrangian actions, enhancing understanding of optimal maps.

Contribution

It introduces new, straightforward criteria for identifying c-convex functions in the context of optimal transport with Lagrangian-based costs.

Findings

Derived simple conditions for c-convexity of smooth functions.

Clarified the structure of optimal maps under Lagrangian costs.

Enhanced theoretical understanding of optimal transportation potentials.

Abstract

Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.