Non-existence of optimal transport maps for the multi-marginal repulsive harmonic cost

TL;DR

This paper provides a counterexample showing that for certain multi-marginal optimal transport problems with a harmonic repulsive cost, no optimal transport map exists that is supported on a graph over the first variable.

Contribution

It demonstrates the non-existence of optimal transport maps in multi-marginal problems with harmonic repulsive costs, challenging assumptions of map existence in such settings.

Findings

Counterexample for the existence of optimal transport maps

No minimizer supported on a graph over the first variable

Applicable for any dimension d ≥ 1

Abstract

We give an example of an absolutely continuous measure $\mu$ on $\mathbb R^d$, for any $d \ge 1$, such that no minimizer of the $3$-marginal harmonic repulsive cost with all marginals equal to $\mu$ is supported on a graph over the first variable.