A genus-zero surface with bounded curvature enclosing less volume than the unit sphere

Abstract

We produce a family of bodies in ${\mathbb{R}}^3$ parameterized by $\epsilon >0$, each bounded by a smooth topological sphere with principal curvatures in $[-1,1]$, and having volume arbitrarily close to $16-4\sqrt{3}+\left(10\sqrt{3}-14\right)\ddot{I}€-\left(\frac{10}{3}-\sqrt{3}\right)\ddot{I}{€}^2\approx \mathrm{3.70.}$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4}{3}\ddot{I}€\approx 4.19$) does not enclose the minimal volume among all smooth spheres in ${\mathbb{R}}^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin.