Structural results on convexity relative to cost functions

TL;DR

This paper explores the concept of cost convexity in mass transportation problems, establishing connections with classical convexity and generalizing Jensen's inequality to deepen understanding of convex potentials in optimization.

Contribution

It introduces a generalized Jensen's inequality for cost convex functions and analyzes their relationship with traditional convexity, advancing theoretical foundations in mass transportation and optimization.

Findings

Established a generalized Jensen's inequality for cost convex functions

Linked cost convexity with classical convexity concepts

Provided theoretical insights into convex potentials in mass transportation

Abstract

Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen's inequality for cost convex functions.