Approximation by BV-extension Sets via Perimeter Minimization in Metric Spaces

We show that every bounded domain in a metric measure space can be approximated in measure from inside by closed $BV$-extension sets. The extension sets are obtained by minimizing the sum of the perimeter and the measure of the difference between the domain and the set. By earlier results, in PI spaces the minimizers have open representatives with locally quasiminimal surface. We give an example in a PI space showing that the open representative of the minimizer need not be a $BV$-extension domain nor locally John.