A new Federer-type characterization of sets of finite perimeter in metric spaces

Abstract

Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here the measure-theoretic boundary $\partial^*E$ consists of those points where both $E$ and its complement have positive upper density. We show that the characterization remains true if $\partial^*E$ is replaced by a smaller boundary consisting of those points where the \emph{lower} densities of both $E$ and its complement are at least a given number. This result is new even in Euclidean spaces but we prove it in a more general complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality.