On Sudakov’s type decomposition of transference plans with norm costs

Abstract

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm costj j D min Z jT(x) xjD d (x); T : R d ! R d ; = T# ; with , probability measures in R d and absolutely continuous w.r.t. L d . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in Za R d , where fZaga2A R d are disjoint regions such that the construction of an optimal map Ta : Za! R d is simpler than in the original problem, and then to obtain T by piecing together the maps Ta. When the normj j D is strictly convex (25), the sets Za are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Ta is straightforward provided one can show that the disintegration ofL d (and thus of ) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdor measure (12). When the normjj D is not strictly convex, the main problems in this kind of approach are two: rst, to identify a suitable family of regionsfZaga2A on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we show how these diculties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set Za and then in R d . The strategy is suciently powerful to be applied to other optimal transportation problems. The analysis requires