Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio, L., Capasso, V., Villa, E.: On the approximation of geometric densities of random closed sets. Available at http://cvgmt.sns.it (2006, preprint)
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)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford (2000)
Aubin J.P.: Mutational equations in metric spaces. Set-Valued Anal. 1, 3–46 (1993)
Capasso V., Villa E.: On mean densities of inhomogeneous geometric processes arising in material science and medicine. Image Anal. Stereol. 26, 23–36 (2007)
Colesanti A., Hug D.: Hessian measures of semi-convex functions and applications to support measures of convex bodies. Manuscripta Math. 101, 209–238 (2000)
Federer H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Federer H.: Geometric Measure Theory. Springer, Berlin (1969)
Hug D., Last G., Weil W.: A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272 (2004)
Kiderlen M., Rataj J.: On infinitesimal increase of volumes of morphological transforms. Mathematika 53, 103–127 (2006)
Lorenz T.: Set-valued maps for image segmentation. Comput. Vis. Sci. 4, 41–57 (2001)
Rataj J.: On boundaries of unions of sets with positive reach. Beiträge Algebra Geom. 46, 397–404 (2005)
Schneider R.: Curvature measures of convex bodies. Ann. Mat. Pura e Appl. 116, 101–134 (1978)
Sokolowski, J., Zolesio, J.P.: Introduction to shape optimization. Shape sensitivity analysis. Springer Series in Computational Mathematics, 16, Springer, Berlin (1992)
Stoyan D., Kendall W.S., Mecke J.: Stochastic Geometry and its Application. Wiley, Chichester (1995)
Zähle M.: Integral and current representation of Federer’s curvature measures. Arch. Math. 46, 557–567 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ambrosio, L., Colesanti, A. & Villa, E. Outer Minkowski content for some classes of closed sets. Math. Ann. 342, 727–748 (2008). https://doi.org/10.1007/s00208-008-0254-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-008-0254-z