Calculus of Variations and Geometric Measure Theory

J. M. Morel - F. Santambrogio

Comparison of distances between measures

created by santambro on 19 Feb 2006
modified on 14 Jan 2009


Published Paper

Inserted: 19 feb 2006
Last Updated: 14 jan 2009

Journal: Appl. Math. Lett.
Volume: 20
Number: 4
Pages: 427-432
Year: 2007


Recent works on optimal transport problems (mainly by Q. Xia who generalised classical discrete approaches to the continuous setting) where it is convenient to keep masses together during the transportation define new distances between probability measures induced by the infima of suitable variational problems. The question arises of how these distances compare to the classical Wasserstein distance obtained by the Monge-Kantorovich problem. In this short paper we show sharp inequalities between the $d_\alpha$ distance induced by branching transport paths and the classical Wasserstein distance over probability measures in a compact domain of $R^m$.

Keywords: Branched transport, Wasserstein distances, sharp inequalities