Rectifiability of Optimal Transportation Plans

TL;DR

This paper proves that optimal transportation plans are supported on Lipschitz manifolds under certain smoothness conditions and uses this to show solutions to Monge's problem satisfy a change of variables almost everywhere.

Contribution

It establishes the rectifiability of optimal transportation plans under $C^{2}$ cost functions with non-singular mixed second derivatives, providing a new proof for Monge's problem solutions.

Findings

Optimal plans are supported on Lipschitz manifolds.

Solutions to Monge's problem satisfy a change of variables almost everywhere.

The result applies to costs with non-singular mixed second derivatives.

Abstract

The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.