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arXiv · 2406.18428

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Andrii Arman·Andriy Bondarenko·Andriy Prymak·Danylo Radchenko
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Abstract

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 $0.891$.

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Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko. 2024-06-26. Small Volume Bodies of Constant Width with Tetrahedral Symmetries. https://doi.org/10.3842/sigma.2025.109

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