6533b7cefe1ef96bd1257041

RESEARCH PRODUCT

Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

Tapio Rajala

subject

Mathematics - Differential GeometryPure mathematicsGeodesicPoincaré inequalityMetric measure spaceCurvature01 natural sciencesConvexitysymbols.namesakeMathematics - Analysis of PDEsMathematics - Metric GeometryFOS: MathematicsMathematics::Metric Geometry0101 mathematicsRicci curvatureMathematicsProbability measure010102 general mathematicsta111Measure contraction propertyMetric Geometry (math.MG)53C23 (Primary) 28A33 49Q20 (Secondary)Functional Analysis (math.FA)010101 applied mathematicsMathematics - Functional AnalysisMetric spaceRicci curvatureDifferential Geometry (math.DG)Poincaré inequalityBounded functionsymbolsMathematics::Differential GeometryAnalysisAnalysis of PDEs (math.AP)

description

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the geodesic. Using these geodesics we show that a local Poincar\'e inequality and the measure contraction property follow from the Ricci curvature bounds defined by Sturm. We also show for a large class of convex functionals that a local Poincar\'e inequality is implied by the weak displacement convexity of the functional.

yearjournalcountryeditionlanguage
2011-11-23
http://arxiv.org/abs/1111.5526