On deterministic solutions for multi-marginal optimal transport with Coulomb cost

TL;DR

This paper investigates the three-marginal optimal transport problem with Coulomb cost in the plane, analyzing the optimality of the Seidl map and providing conditions and counterexamples related to particle alignment.

Contribution

It generalizes previous partial results, establishes a necessary and sufficient condition for cost equivalence, and constructs infinite counterexamples to the Seidl map's optimality.

Findings

Necessary and sufficient condition for Coulomb cost to match aligned particle cost

Disproof of Seidl map's universal optimality with counterexamples

Extension of previous partial positive results on Coulomb optimal transport

Abstract

In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $\R^2$. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.