Characterization of optimal Transport Plans for the Monge-Kantorovich-Problem

TL;DR

This paper proves that c-cyclically monotone transport plans optimize the Monge-Kantorovich problem under a broad measurability condition, removing the need for regularity assumptions previously required.

Contribution

It establishes that c-cyclically monotone plans are optimal under minimal measurability conditions, extending prior results without regularity constraints.

Findings

c-cyclically monotone plans are optimal under measurability conditions

The measurability condition is always satisfied for finitely valued, lower semi-continuous costs

Addresses a problem posed in Villani's book

Abstract

We prove that $c$-cyclically monotone transport plans $\pi$ optimize the Monge-Kantorovich transportation problem under an additional measurability condition. This measurability condition is always satisfied for finitely valued, lower semi-continuous cost functions. In particular, this yields a positive answer to Problem 2.25 in C. Villani's book. We emphasize that we do not need any regularity conditions as were imposed in the previous literature.