A Bishop type inequality on metric measure spaces with Ricci curvature bounded below

2016/03/14 by Yu Kitabeppu, Kitabeppu, Yu
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1603.04162

openalex publication_date 2016/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any RCD spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such RCD^*(K,N) spaces are exactly N. We also prove that every tangent cone at any point on such RCD spaces is a metric cone.

Citations

• A Note on Lipschitz Continuity of Solutions of Poisson Equations in\n Metric Measure Spaces
• Ricci curvature for metric-measure spaces via optimal transport
• On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• The splitting theorem in non-smooth context
• Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

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