Mathematische Annalen

, Volume 342, Issue 4, pp 727–748 | Cite as

Outer Minkowski content for some classes of closed sets

  • Luigi AmbrosioEmail author
  • Andrea Colesanti
  • Elena Villa


We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in \({\mathbb{R}^d}\) , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.

Mathematics Subject Classification (2000)

28A75 49Q15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ambrosio, L., Capasso, V., Villa, E.: On the approximation of geometric densities of random closed sets. Available at (2006, preprint)
  2. 2.
    Ambrosio, L., Dancer, N.: Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. In: Buttazzo, G., Marino, A., Murthy, M.K.V., (eds.) Springer, Berlin (2000)Google Scholar
  3. 3.
    Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)zbMATHGoogle Scholar
  4. 4.
    Aubin J.P.: Mutational equations in metric spaces. Set-Valued Anal. 1, 3–46 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Capasso V., Villa E.: On mean densities of inhomogeneous geometric processes arising in material science and medicine. Image Anal. Stereol. 26, 23–36 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Colesanti A., Hug D.: Hessian measures of semi-convex functions and applications to support measures of convex bodies. Manuscripta Math. 101, 209–238 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Federer H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Federer H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  9. 9.
    Hug D., Last G., Weil W.: A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kiderlen M., Rataj J.: On infinitesimal increase of volumes of morphological transforms. Mathematika 53, 103–127 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lorenz T.: Set-valued maps for image segmentation. Comput. Vis. Sci. 4, 41–57 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rataj J.: On boundaries of unions of sets with positive reach. Beiträge Algebra Geom. 46, 397–404 (2005)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Schneider R.: Curvature measures of convex bodies. Ann. Mat. Pura e Appl. 116, 101–134 (1978)zbMATHCrossRefGoogle Scholar
  14. 14.
    Sokolowski, J., Zolesio, J.P.: Introduction to shape optimization. Shape sensitivity analysis. Springer Series in Computational Mathematics, 16, Springer, Berlin (1992)Google Scholar
  15. 15.
    Stoyan D., Kendall W.S., Mecke J.: Stochastic Geometry and its Application. Wiley, Chichester (1995)Google Scholar
  16. 16.
    Zähle M.: Integral and current representation of Federer’s curvature measures. Arch. Math. 46, 557–567 (1986)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di MatematicaUniversità degli Studi di FirenzeFirenzeItaly
  3. 3.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly

Personalised recommendations