Rigidity for measurable sets

2021/02/22 by Dorin Bucur, Bucur, Dorin, Ilaria Fragalà +1
Mathematics · Computer Science · #Point processes and geometric inequalities #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · doi:10.48550/arxiv.2102.12389

Abstract

Let Ω⊂ ℝ^d be a set with finite Lebesgue measure such that, for a fixed radius r>0, the Lebesgue measure of Ω∩ B_r (x) is equal to a positive constant when x varies in the essential boundary of Ω. We prove that Ω is a ball (or a finite union of equal balls) provided it satisfies a nondegeneracy condition, which holds in particular for any set of diameter larger than r which is either open and connected, or of finite perimeter and indecomposable. The proof requires reinventing each step of the moving planes method by Alexandrov in the framework of measurable sets.

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