Skip to main content

A New Federer-Type Characterization of Sets of Finite Perimeter


Federer’s characterization states that a set \(E\subset \mathbb {R}^n\) is of finite perimeter if and only if \(\mathcal H^{n-1}(\partial ^*E)<\infty \). Here the measure-theoretic boundary \(\partial ^*E\) consists of those points where both E and its complement have positive upper density. We show that the characterization remains true if \(\partial ^*E\) is replaced by a smaller boundary consisting of those points where the lower densities of both E and its complement are at least a given number. This result is new even in Euclidean spaces but we prove it in a more general complete metric space that is equipped with a doubling measure and supports a Poincaré inequality.

This is a preview of subscription content, access via your institution.


  1. 1.

    Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces, calculus of variations, nonsmooth analysis and related topics. Set Valued Anal. 10(2–3), 111–128, 2002

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ambrosio, L., Caselles, V., Masnou, S., Morel, J.-M.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. (JEMS)3(1), 39–92, 2001

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, New York 2000

    MATH  Google Scholar 

  4. 4.

    Ambrosio, L., Miranda Jr., M., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces. In: Pallara, D. (ed.) Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi. Quaderni di matematica, vol. 14, pp. 1–45. Department of Mathematics, University of Napoli, Caserta 2004

    Google Scholar 

  5. 5.

    Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, vol. 17. European Mathematical Society (EMS), Zürich 2011

    Book  Google Scholar 

  6. 6.

    Buckley, S.M.: Is the maximal function of a Lipschitz function continuous? Ann. Acad. Sci. Fenn. Math. 24(2), 519–528, 1999

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Chlebík, M.: Going beyond variation of sets. Nonlinear Anal. 153, 230–242, 2017

    MathSciNet  Article  Google Scholar 

  8. 8.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics Series. CRC Press, Boca Raton 1992

    MATH  Google Scholar 

  9. 9.

    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York 1969

    Google Scholar 

  10. 10.

    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel 1984

    Book  Google Scholar 

  11. 11.

    Hajłasz, P.: Sobolev spaces on metric-measure spaces. In: Auscher, P., Coulhon, T., Grigoryan, A. (eds.) Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemporary Mathematics, vol. 338, pp. 173–218. American Mathematical Society, Providence, RI 2003

    Google Scholar 

  12. 12.

    Hakkarainen, H., Kinnunen, J.: The BV-capacity in metric spaces. Manuscr. Math. 132(1–2), 51–73, 2010

    MathSciNet  Article  Google Scholar 

  13. 13.

    Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York 2001

    Book  Google Scholar 

  14. 14.

    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61, 1998

    MathSciNet  Article  Google Scholar 

  15. 15.

    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev Spaces on Metric Measure Spaces. An Approach Based on Upper Gradients. New Mathematical Monographs, vol. 27. Cambridge University Press, Cambridge 2015

    Book  Google Scholar 

  16. 16.

    Järvenpää, E., Järvenpää, M., Rogovin, K., Rogovin, S., Shanmugalingam, N.: Measurability of equivalence classes and \(MEC_p\)-property in metric spaces. Rev. Mat. Iberoam. 23(3), 811–830, 2007

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kelly, J.C.: Quasiconformal mappings and sets of finite perimeter. Trans. Am. Math. Soc. 180, 367–387, 1973

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kinnunen, J., Korte, R., Lorent, A., Shanmugalingam, N.: Regularity of sets with quasiminimal boundary surfaces in metric spaces. J. Geom. Anal. 23(4), 1607–1640, 2013

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Pointwise properties of functions of bounded variation in metric spaces. Rev. Mat. Complut. 27(1), 41–67, 2014

    MathSciNet  Article  Google Scholar 

  20. 20.

    Lahti, P.: A Federer-style characterization of sets of finite perimeter on metric spaces. Calc. Var. Partial Differ. Equ. 56(5), 22, 2017

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lahti, P.: A sharp Leibniz rule for BV functions in metric spaces. Rev. Mat. Complut. arxiv:1811.07713(to appear)

  22. 22.

    Lahti, P.: Capacities and 1-strict subsets in metric spaces. Nonlinear Anal.(to appear)

  23. 23.

    Lahti, P.: Federer’s characterization of sets of finite perimeter in metric spaces. Anal. PDE. arxiv:1804.11216(to appear)

  24. 24.

    Lahti, P.: Strong approximation of sets of finite perimeter in metric spaces. Manuscr. Math. 155(3–4), 503–522, 2018

    MathSciNet  Article  Google Scholar 

  25. 25.

    Lahti, P.: Superminimizers and a weak Cartan property for \(p=1\) in metric spaces. J. Anal. Math. arxiv:1706.01873(to appear)

  26. 26.

    Lorent, A.: On indecomposable sets with applications. ESAIM Control Optim. Calc. Var. 20(2), 612–631, 2014

    MathSciNet  Article  Google Scholar 

  27. 27.

    Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9)82(8), 975–1004, 2003

    MathSciNet  Article  Google Scholar 

  28. 28.

    Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York 1988

    MATH  Google Scholar 

  29. 29.

    Shanmugalingam, N.: Harmonic functions on metric spaces. Ill. J. Math. 45(3), 1021–1050, 2001

    MathSciNet  Article  Google Scholar 

  30. 30.

    Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16(2), 243–279, 2000

    MathSciNet  Article  Google Scholar 

  31. 31.

    Ziemer, W.P.: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, vol. 120. Springer, New York 1989

    MATH  Google Scholar 

Download references


The author wishes to thank Nageswari Shanmugalingam for many helpful comments as well as for discussions on constructing spaces where the Mazurkiewicz metric agrees with the ordinary one; Anders Björn also for discussions on constructing such spaces; and Olli Saari for discussions on finding strong boundary points.

Author information



Corresponding author

Correspondence to Panu Lahti.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by G. Dal Maso

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lahti, P. A New Federer-Type Characterization of Sets of Finite Perimeter. Arch Rational Mech Anal 236, 801–838 (2020).

Download citation