Calculus of Variations and Geometric Measure Theory

D. Bucur - I. FragalĂ 

Rigidity for measurable sets

created by bucur on 25 Feb 2021



Inserted: 25 feb 2021

Year: 2021

ArXiv: 2102.12389 PDF


Let $\Omega \subset \mathbb R^d$ be a set with finite Lebesgue measure such that, for a fixed radius $r>0$, the Lebesgue measure of $\Omega \cap B _ r (x)$ is equal to a positive constant when $x$ varies in the essential boundary of $\Omega$. We prove that $\Omega$ 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.