Small volume bodies of constant width with tetrahedral symmetries

Overview & Synopsis

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 $0.891$.