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 cost $|\cdot {|}_{D^{\ast }}$ \[ \min \bigg\{\int |\mathtt T(x) - x|_{D^*} dμ(x), \ \mathtt T : \mathbb R^d \to \mathbb R^d, \ ν= \mathtt T_\# μ\bigg\}, \] with $\mu$, $\nu$ probability measures in ${\mathbb{R}}^d$ and $\mu$ absolutely continuous w.r.t. ${\mathcal{L}}^d$. 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 $Z_{\mathfrak{a}}\times {\mathbb{R}}^d$, where $\{Z_{\mathfrak{a}}{\}}_{\mathfrak{a}\in \mathfrak{A}}\subset {\mathbb{R}}^d$ are disjoint regions such that the construction of an optimal map ${\mathtt{T}}_{\mathfrak{a}}:Z_{\mathfrak{a}}\to {\mathbb{R}}^d$ is simpler than in the original problem, and then to obtain $\mathtt{T}$ by piecing together the maps ${\mathtt{T}}_{\mathfrak{a}}$. In this paper we show how 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 $Z_{\mathfrak{a}}$ and then in ${\mathbb{R}}^d$. The strategy is sufficiently powerful to be applied to other optimal transportation problems.

83 pages, 16 figures

Cited by in corpus (1)

• A characterization for solutions of the Monge-Kantorovich mass transport problem