Small volume bodies of constant width

Abstract

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n\text{Vol}({\mathbb{B}}^n$), where ${\mathbb{B}}^n$ is the unit ball in ${\mathbb{R}}^n$. This answers a question of O.~Schramm.

8 pages, 4 figures