Abstract
We prove that if \((X,\mathsf {d},\mathfrak {m})\) is a metric measure space with \(\mathfrak {m}(X)=1\) having (in a synthetic sense) Ricci curvature bounded from below by \(K>0\) and dimension bounded above by \(N\in [1,\infty )\), then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counterparts proved in the smooth setting by Milman for any \(K\in \mathbb {R}\), \(N\ge 1\) and upper diameter bounds) holds, i.e. the isoperimetric profile function of \((X,\mathsf {d},\mathfrak {m})\) is bounded from below by the isoperimetric profile of the model space. Moreover, if equality is attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions. Finally we also establish the almost rigidity: if the equality is almost attained for some volume \(v \in (0,1)\) and K is strictly positive, then the space must be mGH close to a spherical suspension. To our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying Ricci curvature lower bounds. Examples of spaces fitting our assumptions include measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.
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Notes
During all the paper we will assume \((X,\mathsf {d})\) to be complete, separable and proper
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Acknowledgements
The authors wish to thank Emanuel Milman for having drawn their attention to the recent paper by Klartag [44] and the reviewers, whose detailed comments led to an improvement of the manuscript. They also wish to thank the Hausdorff center of Mathematics of Bonn, where most of the work has been developed, for the excellent working conditions and the stimulating atmosphere during the trimester program “Optimal Transport” in Spring 2015. The second author gratefully acknowledges the support of the ETH-fellowship.
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Cavalletti, F., Mondino, A. Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds. Invent. math. 208, 803–849 (2017). https://doi.org/10.1007/s00222-016-0700-6
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DOI: https://doi.org/10.1007/s00222-016-0700-6