  We prove existence of an optimal transport map in the Monge-Kantorovich
problem associated to a cost $c(x,y)$ which is not finite everywhere, but
coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex
set $C$ and it is $+\infty$ otherwise. The result is proven for $C$ satisfying
some technical assumptions allowing any convex body in $\R^2$ and any convex
polyhedron in $\R^d$, $d>2$. The tools are inspired by the recent
Champion-DePascale-Juutinen technique. Their idea, based on density points and
avoiding disintegrations and dual formulations, allowed to deal with $L^\infty$
problems and, later on, with the Monge problem for arbitrary norms.
