The paper reviews and partly re-proves results showing that a doubling metric measure space is a PI space if and only if its separating sets have enough weighted boundary energy.
Fine properties of sets of finite perimeter in doubling metric measure spaces
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Geometric characterizations of ${\sf PI}$ spaces: an overview of some modern techniques
The paper reviews and partly re-proves results showing that a doubling metric measure space is a PI space if and only if its separating sets have enough weighted boundary energy.