A general condition for Monge solutions in the multi-marginal optimal transport problem

TL;DR

This paper establishes a broad condition on cost functions that guarantees Monge solutions and uniqueness in multi-marginal optimal transport, unifying and extending existing results and offering a method to generate new examples.

Contribution

It introduces a general sufficient condition on cost functions for Monge solutions and uniqueness, unifying various prior results and providing a systematic way to create new examples.

Findings

Identifies a general condition ensuring Monge solutions.

Proves the condition guarantees uniqueness of solutions.

Provides a method to generate new examples from existing ones.

Abstract

We develop a general condition on the cost function which is sufficient to imply Monge solution and uniqueness results in the multi-marginal optimal transport problem. This result unifies and generalizes several results in the rather fragmented literature on multi-marginal problems. We also provide a systematic way to generate new examples from old ones.