Characterizing unit spheres in Euclidean spaces via reach and volume

Abstract

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