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

Аннотация: 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 $0.891$.

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

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

Поступила: 4 июня 2025 г.; в окончательном варианте 6 декабря 2025 г.; опубликована 21 декабря 2025 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2025.109

ArXiv: 2406.18428