On the Structure of Optimal Transportation Plans between Discrete Measures

Abstract

AbstractIt is well known that the optimal transportation plan between two probability measures $$\mu $$ μ and $$\nu $$ ν is induced by a transportation map whenever $$\mu $$ μ is an absolutely continuous measure supported over a compact set in the Euclidean space and the cost function is a strictly convex function of the Euclidean distance. However, when $$\mu $$ μ and $$\nu $$ ν are both discrete, this result is generally false. In this paper, we prove that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as the sum of two deterministic plans, i.e., plans induced by transportation maps. As an application, we estimate the infinity-Wasserstein distance between two discrete probability measures $$\mu $$ μ and $$\nu $$ ν with the p-Wasserstein distance, times a constant depending on $$\mu $$ μ , on $$\nu $$ ν , and on the fixed cost function.