Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data

TL;DR

This paper disproves a conjecture in Density Functional Theory related to multimarginal optimal transport with Coulomb cost and provides examples of optimal maps for specific data classes.

Contribution

It introduces counterexamples to a conjecture and constructs optimal transport maps for particular symmetric data cases.

Findings

Disproved a key conjecture in the field.

Provided explicit examples of optimal maps.

Enhanced understanding of multimarginal optimal transport with Coulomb cost.

Abstract

We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. We also provide examples of maps satisfying optimality conditions for special classes of data.