SIGMA 21 (2025), 109, 8 pages      arXiv:2406.18428      https://doi.org/10.3842/SIGMA.2025.109

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Andrii Arman ^a, Andriy Bondarenko ^b, Andriy Prymak ^a and Danylo Radchenko ^c
a) Department of Mathematics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada
^b) Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
^c) Université de Lille, CNRS, Laboratoire Paul Painlevé, F-59655 Villeneuve d'Ascq, France

Received June 04, 2025, in final form December 06, 2025; Published online December 21, 2025

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 $0.891$.

Key words: bodies of constant width; tetrahedral symmetry; Meissner's bodies.

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