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Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ withoutboundary 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 $\mathbb{S}^k$.

Characterizing unit spheres in Euclidean spaces via reach and volume