Journal
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Volume 50, Issue 3-4, Pages 831-846Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00526-013-0657-x
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Funding
- European Project ERC AdG *GeMe ThNES*
- Academy of Finland [137528]
- Academy of Finland (AKA) [137528, 137528] Funding Source: Academy of Finland (AKA)
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We prove that in metric measure spaces where the entropy functional is -convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one optimal transport plan between any two absolutely continuous measures with finite second moments and this plan is given by a map. The results are applicable in metric measure spaces having Riemannian Ricci curvature bounded below, and in particular they hold also for Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below by some constant.
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