Small Volume Bodies of Constant Width with Tetrahedral Symmetries
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915688880275456 |
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| author | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| author_facet | Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo |
| contents | For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Volume Bodies of Constant Width with Tetrahedral Symmetries Arman, Andrii Bondarenko, Andriy Prymak, Andriy Radchenko, Danylo Metric Geometry 52A20, 52A15, 52A23, 52A40, 28A75, 49Q20 For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$. |
| title | Small Volume Bodies of Constant Width with Tetrahedral Symmetries |
| topic | Metric Geometry 52A20, 52A15, 52A23, 52A40, 28A75, 49Q20 |
| url | https://arxiv.org/abs/2406.18428 |