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

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.

13 pages, 2 figures