*Accepted Paper*

**Inserted:** 26 may 2011

**Last Updated:** 14 oct 2011

**Journal:** J. Math. Pures et Appl.

**Year:** 2011

**Abstract:**

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 $\mathbb{R}^2$ and any convex polyhedron in $\mathbb{R}^d$, $d>2$. 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.

**Keywords:**
optimal transportation, convex bodies, existence of optimal transport maps, c-monotonicity

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