# Capacities and 1-strict subsets in metric spaces

## TL;DR

This paper investigates the properties of strict subsets with finite variational capacity in complete metric spaces with doubling measures and Poincaré inequalities, focusing on their relation to sets of finite perimeter and applications to BV functions.

## Contribution

It characterizes strict subsets that are of finite perimeter using fine topology and applies this to condensers, BV capacities, and approximation of BV functions.

## Key findings

- Characterization of strict subsets of finite perimeter
- Application to condensers and BV capacities
- Pointwise approximation results for BV functions

## Abstract

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$. Relying on the concept of fine topology, we give a characterization of those strict subsets that are also sets of finite perimeter, and then we apply this to the study of condensers as well as BV capacities. We also apply the theory to prove a pointwise approximation result for functions of bounded variation.