The Journal of Geometric Analysis

, Volume 26, Issue 4, pp 2777–2796 | Cite as

Measure-Theoretic Properties of Level Sets of Distance Functions

  • Daniel Kraft


We consider the level sets of distance functions from the point of view of geometric measure theory. This lays the foundation for further research that can be applied, among other uses, to the derivation of a shape calculus based on the level-set method. Particular focus is put on the \((n-1)\)-dimensional Hausdorff measure of these level sets. We show that, starting from a bounded set, all sub-level sets of its distance function have finite perimeter. Furthermore, if a uniform-density condition is satisfied for the initial set, one can even show an upper bound for the perimeter that is uniform for all level sets. Our results are similar to existing results in the literature, with the important distinction that they hold for all level sets and not just almost all. We also present an example demonstrating that our results are sharp in the sense that no uniform upper bound can exist if our uniform-density condition is not satisfied. This is even true if the initial set is otherwise very regular (i.e., a bounded Caccioppoli set with smooth boundary).


Geometric measure theory Level set Distance function Hausdorff measure Perimeter Caccioppoli set 

Mathematics Subject Classification

28A75 49Q10 49Q12 



The author would like to thank Wolfgang Ring of the University of Graz for thorough proofreading of the manuscript. This work is supported by the Austrian Science Fund (FWF) and the International Research Training Group IGDK 1754.


  1. 1.
    Alberti, G., Bianchini, S., Crippa, G.: Structure of level sets and Sard-type properties of Lipschitz maps. Ann. della Scuola Normale Superiore di Pisa—Classe Sci. 12(4), 863–902 (2013)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Almgren, F., Taylor, J.E., Wang, L.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387–437 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, 1st edn. Oxford Science, Oxford (2000)zbMATHGoogle Scholar
  4. 4.
    Bucur, D., Zolésio, J.-P.: Free boundary problems and density perimeter. J. Differ. Equ. 126, 224–243 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burger, M.: A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5, 301–329 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Burger, M., Matevosyan, N., Wolfram, M.-T.: A level set based shape optimization method for an elliptic obstacle problem. Math. Models Method Appl. Sci. 21(4), 619–649 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caraballo, D.G.: Areas of level sets of distance functions induced by asymmetric norms. Pac. J. Math. 218(1), 37–52 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries. Advances in Design and Control, 2nd edn. SIAM, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Delfour, M.C., Zolésio, J.-P.: The new family of cracked sets and the image segmentation problem revisited. Commun. Inf. Syst. 4(1), 29–52 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Droske, M., Ring, W.: A Mumford-Shah level-set approach for geometric image registration. SIAM J. Appl. Math. 66(6), 2127–2148 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  12. 12.
    Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  13. 13.
    Kraft, D.: A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis. Interfaces Free Bound. Preprint IGDK-2015-18.
  14. 14.
    Kraft, D.: A Hopf–Lax formula for the level-set equation and applications to PDE-constrained shape optimisation. In: Proceedings of the 19th International Conference on Methods and Models in Automation and Robotics, pp. 498–503. IEEE Xplore (2014)Google Scholar
  15. 15.
    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yeh, J.: Real Analysis: Theory of Measure and Integration, 2nd edn. World Scientific, Singapore (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2015

Authors and Affiliations

  1. 1.Institute of Mathematics, NAWI GrazUniversity of GrazGrazAustria

Personalised recommendations