Some Remarks on Uniqueness and Regularity of Cheeger Sets

Abstract

We show that generically the subsets of ${\mathbb{R}}^N$ with finite volume have a unique Cheeger set, in the sense that there always exists a nearby set which has a unique Cheeger set. We also prove that Cheeger sets are $C^{1,1}$, when the ambient set is $C^{1,1}$.