A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911227155841024 |
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| author | Brunel, Victor-Emmanuel Ohta, Shin-ichi Serres, Jordan |
| author_facet | Brunel, Victor-Emmanuel Ohta, Shin-ichi Serres, Jordan |
| contents | We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $Ω$ and the Busemann function associated with any straight line, the volume of the intersection of $Ω$ and any sublevel set of the Busemann function that contains a barycenter of $Ω$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_15030 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces Brunel, Victor-Emmanuel Ohta, Shin-ichi Serres, Jordan Metric Geometry Probability Statistics Theory We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $Ω$ and the Busemann function associated with any straight line, the volume of the intersection of $Ω$ and any sublevel set of the Busemann function that contains a barycenter of $Ω$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied. |
| title | A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces |
| topic | Metric Geometry Probability Statistics Theory |
| url | https://arxiv.org/abs/2408.15030 |