  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 $\mathbb{S}^k$.
