Spherical coverings and X-raying convex bodies of constant width
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arXiv
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| Format: | Preprint |
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2020
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| author | Bondarenko, A. Prymak, A. Radchenko, D. |
| author_facet | Bondarenko, A. Prymak, A. Radchenko, D. |
| contents | K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray conjecture and the illumination conjecture for convex bodies of constant width in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we give such constructions with fewer than $2^n$ caps for $5\le n\le 15$.
For the illumination number of any convex body of constant width in $\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$ for $n\ge 16$, confirming the above mentioned conjectures for the class of convex bodies of constant width. Thus, our result settles the outstanding cases $7\le n\le 15$.
We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_06398 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spherical coverings and X-raying convex bodies of constant width Bondarenko, A. Prymak, A. Radchenko, D. Metric Geometry Primary 52C17, Secondary 52A20, 52A40, 52C35 K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray conjecture and the illumination conjecture for convex bodies of constant width in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we give such constructions with fewer than $2^n$ caps for $5\le n\le 15$. For the illumination number of any convex body of constant width in $\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$ for $n\ge 16$, confirming the above mentioned conjectures for the class of convex bodies of constant width. Thus, our result settles the outstanding cases $7\le n\le 15$. We also show how to calculate the covering radius of a given discrete point set on the sphere efficiently on a computer. |
| title | Spherical coverings and X-raying convex bodies of constant width |
| topic | Metric Geometry Primary 52C17, Secondary 52A20, 52A40, 52C35 |
| url | https://arxiv.org/abs/2011.06398 |