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 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 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.

Existence of optimal transport maps for crystalline norms

PRATELLI, ALDO
2004

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 R^n, assuming that the initial distribution of mass is absolutely continuous with respect to 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 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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11571/112892
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact