Search results for "measure contraction property"

showing 2 items of 2 documents

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

2011

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.

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)
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Failure of topological rigidity results for the measure contraction property

2014

We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset isometric to $\mathbb{R}$, but does not topologically split. The second space satisfies MCP(2,3) and has diameter $\pi$, which is the maximal possible diameter for a space satisfying MCP(N-1,N), but is not a topological spherical suspension. The latter example gives an answer to a question by Ohta.

Mathematics - Differential Geometrymetric measure spacesGeodesicPhysics::Instrumentation and DetectorsQuantitative Biology::Tissues and Organsmeasure contraction propertyMetric Geometry (math.MG)53C23 (Primary) 28A33 49Q20 (Secondary)Ricci curvature lower boundsTopologyPotential theorymaximal diameter theoremnonbranchingRigidity (electromagnetism)Mathematics - Metric GeometryDifferential Geometry (math.DG)splitting theoremFOS: MathematicsSplitting theoremContraction (operator theory)AnalysisMathematicsgeodesics
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