Insights into capacity-constrained optimal transport.

Abstract

A variant of the classical optimal transportation problem is the following: among all joint measures with fixed marginals and that are dominated by a given density, find the optimal one. Existence and uniqueness of solutions to this variant were established by Korman and McCann. In the present article, we expose an unexpected symmetry leading to explicit examples in two and more dimensions. These are inspired in part by simulations in one dimension that display singularities and topology and in part by two further developments: the identification of all extreme points in the feasible set and an approach to uniqueness based on constructing feasible perturbations.