Convex bodies of constant width with exponential illumination number
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929399674175488 |
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| author | Arman, Andrii Bondarenko, Andriy Prymak, Andriy |
| author_facet | Arman, Andrii Bondarenko, Andriy Prymak, Andriy |
| contents | We show that there exist convex bodies of constant width in $\mathbb{E}^n$ with illumination number at least $(\cos(π/14)+o(1))^{-n}$, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter $1$ in $\mathbb{E}^n$ which cannot be covered by $(2/\sqrt{3}+o(1))^{n}$ balls of diameter $1$, improving a result by J. Bourgain and J. Lindenstrauss. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_10418 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convex bodies of constant width with exponential illumination number Arman, Andrii Bondarenko, Andriy Prymak, Andriy Metric Geometry Combinatorics Primary 52C17, Secondary 52A20, 52A40, 52C35 We show that there exist convex bodies of constant width in $\mathbb{E}^n$ with illumination number at least $(\cos(π/14)+o(1))^{-n}$, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter $1$ in $\mathbb{E}^n$ which cannot be covered by $(2/\sqrt{3}+o(1))^{n}$ balls of diameter $1$, improving a result by J. Bourgain and J. Lindenstrauss. |
| title | Convex bodies of constant width with exponential illumination number |
| topic | Metric Geometry Combinatorics Primary 52C17, Secondary 52A20, 52A40, 52C35 |
| url | https://arxiv.org/abs/2304.10418 |