Optimal maps for the multidimensional Monge-Kantorovich problem

Abstract

Let μ1,…, μN be Borel probability measures on ℝd. Denote by Γ(μ1,…, μN) the set of all N-tuples T=(T1,…, TN) such that Ti:ℝd ↔ ℝd (i = 1,…, N) are Borel-measurable and satisfy μ1 = μi[V] for all Borel V ⊂ ℝd. The multidimensional Monge-Kantorovich problem investigated in this paper consists of finding S=(S1,…, SN) ∈ Γ(μ1,…, μN) minimizing over the set Γ(μ1, ···, μN). We study the case where the μi's have finite second moments and vanish on (d - 1)-rectifiable sets. We prove existence and uniqueness of optimal maps S when we impose that S1(x) ≡ x and give an explicit form of the maps Si. The result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality. As a consequence, we obtain uniqueness and characterization of an optimal measure for the multidimensional Kantorovich problem. © 1998 John Wiley & Sons, Inc.