Optimal transport of vector measures

TL;DR

This paper develops a theory of optimal transport for vector measures, disproves a conjecture by Klartag, and generalizes Kantorovich--Rubinstein duality to this setting, providing conditions for the conjecture to hold.

Contribution

It introduces a new framework for vector measure transport, disproves a key conjecture, and extends duality theory to this broader context.

Findings

Counterexample to Klartag's conjecture.

Generalization of Kantorovich--Rubinstein duality.

Conditions under which the conjecture holds with absolutely continuous marginals.

Abstract

We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich--Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation.