# Dorin Bucur

Université de Savoie

### 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.