Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910570559569920 |
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| author | Han, Bang-Xian |
| author_facet | Han, Bang-Xian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11570 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature Han, Bang-Xian Metric Geometry Functional Analysis 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. |
| title | Sharp and rigid isoperimetric inequality in metric measure spaces with non-negative Ricci curvature |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2212.11570 |