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