Rigidity for measurable sets

TL;DR

This paper characterizes certain measurable sets with uniform local measure properties as being balls or unions of equal balls, extending classical symmetry results through a novel adaptation of the moving planes method.

Contribution

It introduces a new approach to prove rigidity of measurable sets with uniform measure slices, generalizing classical geometric symmetry results to a measurable setting.

Findings

Sets with uniform measure intersection are balls or unions of equal balls.

The proof adapts the moving planes method to measurable sets.

The result applies to sets with diameter larger than r under certain conditions.

Abstract

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.