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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