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

Approximation by BV-extension sets via perimeter minimization in metric spaces