Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Abstract

For every 𝑛 ≥ 2, we construct a body 𝑈ₙ of constant width 2 in 𝔼ⁿ with small volume and symmetries of a regular 𝑛-simplex. 𝑈₂ is the Reuleaux triangle. To the best of our knowledge, 𝑈₃ 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 𝑈₃ 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 𝑈ₙ is smaller than the volume of the ball of radius 0.891.