Uniqueness and Monge solutions in the multi-marginal optimal transportation problem

TL;DR

This paper investigates the multi-marginal optimal transportation problem, establishing conditions under which solutions are unique and correspond to Monge solutions, thus advancing understanding of solution structures in optimal transport.

Contribution

It proves the existence and uniqueness of solutions to both the Kantorovich and Monge formulations under specific conditions on the cost function and first marginal.

Findings

Solutions to the Kantorovich and Monge problems are unique.

The solution to the relaxed Kantorovich problem induces a Monge solution.

Conditions on the cost function ensure the correspondence between solutions.

Abstract

We study a multi-marginal optimal transportation problem. Under certain conditions on the cost function and the first marginal, we prove that the solution to the relaxed, Kantorovich version of the problem induces a solution to the Monge problem and that the solutions to both problems are unique.