Small volume bodies of constant width

TL;DR

The paper constructs small-volume bodies of constant width in high dimensions, providing explicit examples that answer a longstanding open question in geometric analysis.

Contribution

It offers explicit constructions of bodies of constant width with exponentially small volume in high dimensions, addressing a question posed by O. Schramm.

Findings

Constructed bodies of constant width with volume less than 0.9^n times the unit ball volume

Provided explicit examples for all sufficiently large dimensions

Answered an open problem posed by O. Schramm

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.