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

2011/11/23 by Tapio Rajala, Rajala, Tapio · 4 citations
Mathematics · Medicine · #49Q20 (Secondary) #53C23 (Primary) 28A33 #Analysis of PDEs (math.AP) #Bone health and osteoporosis research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1111.5526

openalex publication_date 2011/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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• Mosco-convergence of Cheeger energies on varying spaces satisfying curvature dimension conditions
• Regularity for quasilinear elliptic equations in metric measure spaces
• Stability of local Riemannian Ricci curvature lower bounds
• Comparison estimates on nonsmooth spaces with integrable Ricci lower bounds via localization

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