Optimal maps in essentially non-branching spaces

Abstract

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.

Final version to appear in Communications in Contemporary Mathematics

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