Optimal Transport with Coulomb cost. Approximation and duality

TL;DR

This paper investigates the duality in multimarginal optimal transport problems with Coulomb cost, establishing approximation methods and existence of solutions, and extending the approach to more general costs.

Contribution

It introduces a discrete approximation approach to prove duality and existence of maximizers for Coulomb cost in multimarginal optimal transport, surpassing classical methods.

Findings

Proved equality of extremal values using discrete approximation.

Established existence of dual maximizers (Kantorovich potentials).

Extended the approach to more general cost functions.

Abstract

We revisit the duality theorem for multimarginal optimal transportation problems. In particular, we focus on the Coulomb cost. We use a discrete approximation to prove equality of the extremal values and some careful estimates of the approximating sequence to prove existence of maximizers for the dual problem (Kantorovich's potentials). Finally we observe that the same strategy can be applied to a more general class of costs and that a classical results on the topic cannot be applied here.