Small volume bodies of constant width

A convex body $K \subset \mathbb{R}^n$ is said to be of constant width 2 if its projection onto any line is a segment of length 2.
An obvious example is the unit ball, but many other bodies of constant width exist. Motivated by Borsuks conjecture,
Schramm asked whether there exist constant width bodies in $\mathbb{R}^n$ of volume exponentially smaller than the unit ball $B_n$.
We construct such a body with volume less than $0.9^n \mathrm{Vol}(B_n)$ for all large $n$, answering Schramm's question. Our method
also provides a new body of constant width 2 in $\mathbb{R}^3$ with interesting properties.
Joint work with A. Arman, F. Nazarov, A. Prymak, and D. Radchenko