Calculus of Variations and Geometric Measure Theory

L. Ambrosio - T. Rajala

Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces

created by ambrosio on 22 Nov 2011
modified by rajala1 on 14 Nov 2012


Accepted Paper

Inserted: 22 nov 2011
Last Updated: 14 nov 2012

Journal: Ann. Mat. Pura Appl.
Year: 2012


We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal maps.

Tags: GeMeThNES
Keywords: optimal transportation, geodesic metric space, non-branching, upper gradient