An exterior optimal transport problem

TL;DR

This paper introduces a new variant of the optimal transport problem with a specific marginal constraint and proves the existence and form of maximizers under certain conditions, including radial, increasing, and coercive cost functions.

Contribution

It formulates a novel optimal transport variant with a constrained second marginal and characterizes maximizers for radial, increasing, and coercive costs.

Findings

Maximizers exist under general assumptions on the cost function.

For radial, increasing, and coercive costs, maximizers are characteristic functions of balls of volume m.

The problem extends classical optimal transport by adding a second marginal constraint.

Abstract

This paper deals with a variant of the optimal transportation problem. Given f $\in$ L 1 (R d , [0, 1]) and a cost function c $\in$ C(R d x R d) of the form c(x, y) = k(y -- x), we minimise $\int$ c d$\gamma$ among transport plans $\gamma$ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 -- f. Denoting by $\Upsilon$(f) the infimum of this problem, we then consider the maximisation problem sup{$\Upsilon$(f) : $\int$ f = m} where m \> 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m.