Calculus of Variations and Geometric Measure Theory

F. Cavalletti - A. Mondino

Optimal maps in essentially non-branching spaces

created by cavallett on 01 Sep 2016
modified on 17 Oct 2016


Accepted Paper

Inserted: 1 sep 2016
Last Updated: 17 oct 2016

Journal: Commun. Contemp. Math.
Year: 2016


In this note we prove that in a metric measure space $(X,d,m)$ verifying the measure contraction property with parameters $K \in \mathbb{R}$ and $1< N< \infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $m$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.