Supersqueezed sphere

TL;DR

This paper presents a counterexample of a smooth, genus-zero surface in three-dimensional space with bounded curvature and smaller enclosed volume than a unit ball, answering a question by Burago and Petrunin.

Contribution

It constructs a specific example of a supersqueezed sphere that challenges previous assumptions about volume and curvature constraints.

Findings

Provides a smooth, genus-zero surface with volume less than a unit ball.

Shows the existence of surfaces with bounded curvature and small volume.

Answers a previously open question negatively.

Abstract

This work pose an example of a smooth closed surface in $\mathbb{R}^3$ which has genus $0$, normal curvatures at most $1$ in absolute value and encloses a volume smaller than the volume of a unit ball. It gives a negative answer to a question asked by Dmitri Burago and Anton Petrunin.