Inserted: 14 may 2003
Last Updated: 4 oct 2005
Journal: Duke Math. J.
We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in $*R*^n$, assuming that the initial distribution of mass is absolutely continuous with respect to Lebesgue measure in $*R*^n$. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having as cost function the Euclidean distance. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in $*R*^3$ 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.
Keywords: Gamma-convergence, Optimal transport maps, Nikodym set, Crystalline norms