Monge problem in metric measure spaces with Riemannian curvature-dimension condition

2013/10/15 by Fabio Cavalletti, Cavalletti, Fabio · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG

paper · pdf · doi:10.48550/arxiv.1310.4036

arxiv created 2013/10/15 · arxiv updated 2013/10/16

Abstract

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition \RCD(K,N), with N < ∞. For the first marginal measure, we assume that μ₀ ≪ m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value. Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

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