On a problem of optimal mixing

TL;DR

This paper studies a generalized optimal transportation problem where the target measure is not fixed, proving existence of solutions and connecting Monge and Kantorovich formulations in Euclidean spaces.

Contribution

It introduces a framework for simultaneous optimal transportation with flexible targets and establishes the equivalence of Monge and Kantorovich solutions in Euclidean spaces.

Findings

Existence of solutions in completely regular spaces.

Equivalence of Monge and Kantorovich problems in Euclidean space.

Structural analysis of the discrete problem.

Abstract

We consider the simultaneous optimal transportation of measures, where the target marginal is not necessarily fixed. For this problem, we prove the existence of a solution for completely regular spaces and investigate the structure of the discrete problem. We establish a connection between the Monge problem and the Kantorovich problem by showing that their functionals are equal and that the solutions coincide in Euclidean space.