The sphere is the only closed surface satisfying the fixed-width Archimedean property

Abstract

Among Archimedes' many celebrated discoveries is a striking fact: the region of the unit sphere between two parallel planes, both meeting the sphere and separated by a distance $h$, has area $2\pi h$. We prove that, among connected smooth closed embedded surfaces in ${\mathbb{R}}^3$, this property characterizes the unit sphere for a single fixed width.

arXiv commentThis paper will be subsumed by an upcoming submission by the author together with Mohammad Ghomi and Ralph Howard where we generalize the results using similar methods.