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Semmes family of curves and a characterization of functions of bounded variation in terms of curves. (English) Zbl 1327.31026

Summary: On metric spaces supporting a geometric version of a Semmes family of curves, we provide a Reshetnyak-type characterization of functions of bounded variation in terms of the total variation on such a family of curves. We then use this characterization to obtain a Federer-type characterization of sets of finite perimeter, that is, we show that a measurable set is of finite perimeter if and only if the Hausdorff measure of its measure theoretic boundary is finite. We present a construction of a geometric Semmes family of curves in the first Heisenberg group.

MSC:

31E05 Potential theory on fractals and metric spaces
30L99 Analysis on metric spaces
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