Characterizing unit spheres in Euclidean spaces via reach and volume

Abstract

Let $M$ be a smooth, connected, compact submanifold of ${\mathbb{R}}^n$ without boundary and of dimension $k\ge 2$. Let ${\mathbb{S}}^k\subset {\mathbb{R}}^{k+1}\subset {\mathbb{R}}^n$ denote the $k$-dimesnional unit sphere. We show if $M$ has reach equal to one, then its volume satisfies $\text{vol}(M)\ge \text{vol}({\mathbb{S}}^k)$ with equality holding only if $M$ is congruent to ${\mathbb{S}}^k$.