### Rigidity results for measurable sets

Analysis and Geometry Seminar

30th September 2021, 3:15 pm – 4:15 pm

Fry Building, 2.04

Let $\Omega \subset \R^d$ 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. This is joint work with Ilaria Fragala.

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