Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Andrii Arman^a, Andriy Bondarenko^b, Andriy Prymak^a, Danylo Radchenko<sup>c

a</sup> 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

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 $0.891$.

Keywords: bodies of constant width, tetrahedral symmetry, Meissner's bodies.

MSC: 52A20, 52A15, 52A23, 52A40, 28A75, 49Q20

Received: June 4, 2025; in final form December 6, 2025; Published online December 21, 2025

Language: English

DOI: 10.3842/SIGMA.2025.109

ArXiv: 2406.18428