0000000001278058

AUTHOR

Karl-theodor Sturm

showing 2 related works from this author

Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below

2013

We prove existence and uniqueness of optimal maps on $RCD^*(K,N)$ spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation.

Mathematics - Differential GeometryExponentiationLower Ricci bounds; Optimal maps; Optimal transport; RCD spaces01 natural sciencesMeasure (mathematics)Combinatoricssymbols.namesakeMathematics - Metric GeometryRCD spacesSettore MAT/05 - Analisi MatematicaFOS: MathematicsOptimal transportMathematics::Metric GeometryUniqueness0101 mathematicsLower Ricci bounds[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG]Ricci curvatureMathematicsDiscrete mathematics010102 general mathematicsMetric Geometry (math.MG)Absolute continuity16. Peace & justice010101 applied mathematicsMathematics::LogicDifferential geometryDifferential Geometry (math.DG)Fourier analysisBounded functionsymbolsOptimal mapsGeometry and Topology
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Non-branching geodesics and optimal maps in strong CD(K,∞) -spaces

2014

We prove that in metric measure spaces where the entropy functional is Kconvex 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. peerReview…

metric measure spacesoptimal mapssMathematics::Metric GeometryMathematics::Differential Geometrynon-branching geodesic
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