**Inserted:** 6 jun 2009

**Last Updated:** 10 sep 2009

**Year:** 2009

**Abstract:**

We establish existence of solutions to the Monge problem in $R^n$ with a norm cost function, assuming absolute continuity of the initial measure.

The proof is based on a one dimensional reduction argument.

The loss in strict convexity implies that transport is possible along several directions. As for crystalline norms, we single out particular solutions to the Kantorovich relaxation with a secondary variational problem involving a strictly convex norm. We then define a map rearranging the mass within the rays, with a one-dimensional Sudakov-type argument proved by a regularity of the disintegration of the Lebesgue measure w.r.t. the rays.

Remark: The construction presently given in the preprint needs the further technical assumption that, with the notation of Section 3.4, the set of points $x$ in $\mathcal{T}_{\mathrm s}$ whose secondary transport ray belongs to the relative border of the convex envelope of $\{y:\phi(x)-\phi(y)=| y-x|\}$ is $\mu$-negligible.\end{abstract}

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