Quantitative isoperimetry à la Levy-Gromov

Abstract

On a Riemannian manifold with a positive lower bound on the Ricci tensor, the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius. The deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise. These results are actually obtained in the more general context of (possibly non-smooth) metric measure spaces with curvature-dimension conditions through a quantitative analysis of the transport-rays decompositions obtained by the localization method.

arXiv comment31 pages. Final version to appear in Comm. Pure Applied Math. From the previous version, we added some more background in the introduction and updated the bibliography

References in corpus (11)

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• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• Alexandrov meets Lott--Villani--Sturm
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• Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality
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Cited by in corpus (7)

• Quantitative Obata's Theorem
• Quantitative isoperimetric inequalities for classical capillarity problems
• Sharp quantitative estimates of Struwe's Decomposition
• Quantitative estimates for the Bakry-Ledoux isoperimetric inequality. II
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• Equality in the logarithmic Sobolev inequality
• The first Dirichlet eigenvalue and the width