Characterizing unit spheres in Euclidean spaces via reach and volume

TL;DR

This paper establishes a geometric characterization of unit spheres in Euclidean spaces by relating the reach and volume of smooth submanifolds, showing that a reach of one implies minimal volume only for spheres.

Contribution

It proves that among smooth, compact submanifolds with reach one, the unit sphere uniquely minimizes volume, providing a new geometric characterization.

Findings

If a submanifold has reach one, its volume is at least that of the unit sphere.

Equality in volume occurs only when the submanifold is congruent to the unit sphere.

The result characterizes spheres via reach and volume in Euclidean spaces.

Abstract

Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ without boundary and of dimension $k\geq 2$. 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 $\text{vol}(M)\geq \text{vol}(\mathbb{S}^k)$ with equality holding only if $M$ is congruent to $\mathbb{S}^k$.