On Sudakov's type decomposition of transference plans with norm costs (1311.1918v2)

Abstract: We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost $|\cdot|{D^*}$ [ \min \bigg{\int |\mathtt T(x) - x|{D^*} d\mu(x), \ \mathtt T : \mathbb R^d \to \mathbb R^d, \ \nu = \mathtt T_# \mu \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.