Characterizing unit spheres in Euclidean spaces via reach and volume

2022/02/12 by Mark Iwen, Iwen, Mark, B. Schmidt +4 · 1 citation
Mathematics · #53Z50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.DG #msc:53Z50

paper · pdf · doi:10.48550/arxiv.2202.06161

arxiv created 2022/02/12 · openalex publication_date 2022/02/12 · arxiv updated 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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