If a Caccioppoli set $A$ is of minimal perimeter in every compact set $K$ contained in some open set, can we say that $A$ is of minimal perimeter in the open set? If not, please construct a counterexample. This problems arises when I read lemma 9.1 in the book of Guisti, Minimal Surfaces and Functions of Bounded Variation. The author explains Caccioppoli sets of least area in this way, and sometimes he uses the term Caccioppoli sets of least perimeter.


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