Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature

2022/12/22 by Bang-Xian Han, Han, Bang-Xian
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2212.11570

openalex publication_date 2022/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

By using optimal transport theory, we prove a sharp dimension-free isoperimetric inequality involving the volume entropy, in metric measure spaces with non-negative Ricci curvature in the sense of Lott--Sturm--Villani. We show that this isoperimetric inequality is attained by a non-trivial open set, if and only if the space satisfies a certain foliation property. For metric measure spaces with non-negative Riemannian Ricci curvature, we show that the sharp Cheeger constant is achieved by a non-trivial measurable set, if and only if a one-dimensional space is split off. Our isoperimetric inequality and the rigidity theorems are proved in non-smooth framework, but new even in the smooth setting. In particular, our results provide some new understanding of logarithmically concave measures.

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