Semmes family of curves and a characterization of functions of bounded variation in terms of curves

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 nite perimeter, that is, we show that a measurable set is of nite perimeter if and only if the Hausdor measure of its measure theoretic boundary is nite. We present a construction of a geometric Semmes family of curves in the rst Heisenberg group.