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Curvature measures and random sets II
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  • Published: January 1986

Curvature measures and random sets II

  • M. Zähle1 

Probability Theory and Related Fields volume 71, pages 37–58 (1986)Cite this article

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Summary

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)

    Measurabilities

  3. 3)

    Geometric behaviour

In the present paper second order local geometric properties of random subsets of R d 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.

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Authors and Affiliations

  1. Sektion Mathematik, UUH, Friedrich-Schiller-Universität Jena, DDR-6900, Jena, German Democratic Republic

    M. Zähle

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  1. M. Zähle
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Zähle, M. Curvature measures and random sets II. Probab. Th. Rel. Fields 71, 37–58 (1986). https://doi.org/10.1007/BF00366271

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  • Received: 02 May 1984

  • Revised: 28 March 1985

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00366271

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Keywords

  • Radon
  • Point Process
  • Invariance Property
  • Radon Measure
  • Geometric Object
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