Existence of optimal transport maps for crystalline norms

Abstract

We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝⁿ, assuming that the initial distribution of mass is absolutely continuous with respect to $\mathcal{L}$ⁿ. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ³ having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.