On a Gallai-type problem and illumination of spiky balls and cap bodies
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arXiv
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| Format: | Preprint |
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2024
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| author | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| author_facet | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| contents | We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Gallai-type problem and illumination of spiky balls and cap bodies Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo Metric Geometry Combinatorics Primary 52A20, Secondary 52A35, 52A37, 52A40, 52C17, 05D15 We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above. |
| title | On a Gallai-type problem and illumination of spiky balls and cap bodies |
| topic | Metric Geometry Combinatorics Primary 52A20, Secondary 52A35, 52A37, 52A40, 52C17, 05D15 |
| url | https://arxiv.org/abs/2408.01341 |