Calculus of Variations and Geometric Measure Theory

P. Pegon - D. Piazzoli - F. Santambrogio

Full characterization of optimal transport plans for concave costs

created by santambro on 13 Nov 2013
modified on 16 Sep 2014


Accepted Paper

Inserted: 13 nov 2013
Last Updated: 16 sep 2014

Journal: Discr. Cont. Dyn. Syst. - A
Volume: special issue
Year: 2014


This paper slightly improves a classical result by Gangbo and McCann (1996) about the structure of optimal transport plans for costs that are concave functions of the Euclidean distance. Since the main difficulty for proving the existence of an optimal map comes from the possible singularity of the cost at $0$, everything is quite easy if the supports of the two measures are disjoint; Gangbo and McCann proved the result under the assumption $\mu(\mathrm{spt}(\nu))=0$; in this paper we replace this assumption with the fact that the two measures are singular to each other. In this case it is possible to prove the existence of an optimal transport map, provided the starting measure $\mu$ does not give mass to small sets (i.e. $(d\!-\!1)-$rectifiable sets). When the measures are not singular the optimal transport plan decomposes into two parts, one concentrated on the diagonal and the other being a transport map between mutually singular measures.

Keywords: Rectifiable sets, Monge-Kantorovich, density points, approximate gradient, transport maps