An improved stability result for Grünbaum's inequality
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908449257816064 |
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| author | Castrillón, Luca Tanganelli |
| author_facet | Castrillón, Luca Tanganelli |
| contents | Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Grünbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$. The inequality is sharp and equality is reached only if $K$ is a cone. Moreover, bodies that almost achieve equality are geometrically close to being cones, as Groemer showed in 2000 by giving his stability estimates for Grünbaum's inequality. In this paper, we improve the exponent in the stability inequality from Groemer's $\frac{1}{2n^2}$ to $\frac{1}{2n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An improved stability result for Grünbaum's inequality Castrillón, Luca Tanganelli Metric Geometry 52A40 Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Grünbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$. The inequality is sharp and equality is reached only if $K$ is a cone. Moreover, bodies that almost achieve equality are geometrically close to being cones, as Groemer showed in 2000 by giving his stability estimates for Grünbaum's inequality. In this paper, we improve the exponent in the stability inequality from Groemer's $\frac{1}{2n^2}$ to $\frac{1}{2n}$. |
| title | An improved stability result for Grünbaum's inequality |
| topic | Metric Geometry 52A40 |
| url | https://arxiv.org/abs/2410.12072 |