Small Volume Bodies of Constant Width with Tetrahedral Symmetries

TL;DR

This paper constructs bodies of constant width with tetrahedral symmetries in any dimension, notably introducing the three-dimensional body $U_3$, which has a smaller volume than previously known bodies with similar symmetry.

Contribution

It provides the first known construction of a three-dimensional body of constant width with tetrahedral symmetry and minimal volume, extending the understanding of such bodies in higher dimensions.

Findings

The body $U_3$ has smaller volume than other 3D bodies with tetrahedral symmetry.

The volume of $U_3$ is within 0.137% of Meissner's bodies of the same width.

For large $n$, $U_n$ has volume less than a sphere of radius 0.891.

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 $0.891$.