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

## TL;DR

This paper extends Federer's finite perimeter characterization to metric spaces by replacing the measure-theoretic boundary with a boundary defined via lower densities, applicable in more general settings.

## Contribution

It introduces a new Federer-type boundary condition based on lower densities, valid in complete metric spaces with doubling measures and Poincaré inequalities.

## Key findings

- The new boundary condition characterizes finite perimeter sets in metric spaces.
- The result generalizes Federer's classical Euclidean theorem.
- The characterization is valid in spaces with doubling measures and Poincaré inequalities.

## 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.