  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.
