On the Many Faces of Easily Covered Polytopes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916449540374528 |
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| author | Florentin, Dan I. Milo, Tomer |
| author_facet | Florentin, Dan I. Milo, Tomer |
| contents | Assume that $rB_{2}^{n} \subset P$ for some polytope $P \subset \mathbb{R}^n$, where $r \in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball $B_2^n$. We prove that if $\log N \le\frac{n}{8}$, then \[ |\mathcal{F}| \ge \left( \frac{1}{ 2\left(1 - r \sqrt{1-\frac{4\log N}{n}}\right) } \right)^{\frac{n-1}{2}}. \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Many Faces of Easily Covered Polytopes Florentin, Dan I. Milo, Tomer Metric Geometry 52A23, 52A37, 52C17, 05B40, 52B11 Assume that $rB_{2}^{n} \subset P$ for some polytope $P \subset \mathbb{R}^n$, where $r \in (\frac{1}{2},1]$. Denote by $\mathcal{F}$ the set of facets of $P$, and by $N=N(P,B_2^n)$ the covering number of $P$ by the Euclidean unit ball $B_2^n$. We prove that if $\log N \le\frac{n}{8}$, then \[ |\mathcal{F}| \ge \left( \frac{1}{ 2\left(1 - r \sqrt{1-\frac{4\log N}{n}}\right) } \right)^{\frac{n-1}{2}}. \] |
| title | On the Many Faces of Easily Covered Polytopes |
| topic | Metric Geometry 52A23, 52A37, 52C17, 05B40, 52B11 |
| url | https://arxiv.org/abs/2410.17811 |