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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 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.

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    We thank Stephen Semmes for pointing out that the initial idea of such a family of curves can be found in the works of Mary Weiss on lacunary series.


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Corresponding author

Correspondence to Nageswari Shanmugalingam.

Additional information

We thank Riikka Kangaslampi for illuminating conversations on the geometry of geodesics in the Bourdon-Pajot spaces. We also thank Luigi Ambrosio and Juha Kinnunen for their encouragement with this project, and the anonymous referee for suggestions that helped improve the exposition of the paper. R. K. was partially supported by Academy of Finland, Grant #250403. P. L. was supported by the Finnish Academy of Science and Letters, the Vilho, Yrjö and Kalle Väisälä Foundation. N. S. was partially supported by NSF Grants DMS-0355027 and DMS-1200915.

Communicated by L. Ambrosio.

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Korte, R., Lahti, P. & Shanmugalingam, N. Semmes family of curves and a characterization of functions of bounded variation in terms of curves. Calc. Var. 54, 1393–1424 (2015).

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Mathematics Subject Classification

  • Primary 31E05
  • Secondary 26A45
  • 26B30
  • 30L99