Probability Theory and Related Fields

, Volume 71, Issue 1, pp 37–58

Curvature measures and random sets II

  • M. Zähle

DOI: 10.1007/BF00366271

Cite this article as:
Zähle, M. Probab. Th. Rel. Fields (1986) 71: 37. doi:10.1007/BF00366271


In choosing models of stochastic geometry three general problems play a role which are closely connected with each other:
  1. 1)

    Construction of the random geometric objects under consideration

  2. 2)


  3. 3)

    Geometric behaviour


In the present paper second order local geometric properties of random subsets of Rd are of interest. These properties are described by signed curvature measures in a measure geometric context.

The theory of point processes on general spaces (here on the space of subsets with positive reach) provides an appropriate framework for solving construction and measurability problems.

Mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties. Ergodic interpretations of the curvature densities are also given.

The appendix provides auxiliary results for random signed Radon measures in locally compact separable Hausdorff spaces.

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • M. Zähle
    • 1
  1. 1.Sektion Mathematik, UUHFriedrich-Schiller-Universität JenaJenaGerman Democratic Republic