The five gradients inequality for non quadratic costs

TL;DR

This paper proves the five gradients inequality in Optimal Transportation Theory for a broad class of cost functions that are smooth, strictly convex, and radially symmetric, extending previous results.

Contribution

It provides a proof of the five gradients inequality for general radially symmetric convex costs, broadening the scope of the inequality's applicability.

Findings

Established the five gradients inequality for a new class of cost functions

Extended the theoretical framework of Optimal Transportation Theory

Provided mathematical tools for analyzing radially symmetric costs

Abstract

We give a proof of the "five gradients inequality" of Optimal Transportation Theory for general costs of the form $c(x,y)=h(x-y)$ where $h$ is a $C^1$ strictly convex radially symmetric function.