An alternative definition for c-convex functions and another synthetic statement of MTW condition

TL;DR

This paper introduces an alternative definition of c-convex functions, proves their equivalence under the MTW condition, and explores related properties, providing a new perspective on optimal transport theory.

Contribution

It establishes the equivalence between c-convexity and an alternative form under the MTW condition, offering a novel synthetic statement of this key condition.

Findings

Proves the equivalence of c-convexity and alternative c-convexity under MTW.

Studies properties of alternative c-convex functions.

Provides a new synthetic formulation of the MTW condition.

Abstract

The main theorem of this paper states that the c-convexity and the alternative c-convexity are equivalent if and only if the cost function c satisfies MTW condition. The alternative c-convex function is an analogy of the definition of the convex function that is using the inequality phi(t x1 + (1 - t) x0) <= t phi(x1) + (1 - t) phi(x0). We study properties of the alternative c-convex functions and MTW condition, then prove the main theorem.