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On the second derivative of the spherical contact distribution function of smooth grain models
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  • Published: September 2001

On the second derivative of the spherical contact distribution function of smooth grain models

  • G. Last1 &
  • R. Schassberger2 

Probability Theory and Related Fields volume 121, pages 49–72 (2001)Cite this article

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  • 8 Citations

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Abstract.

We consider a stationary grain model Ξ in ℝd with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density.

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Authors and Affiliations

  1. Universität Karlsruhe, Institut für Mathematische Stochastik, 76128 Karlsruhe, Germany. e-mail: g.last@math.uni-karlsruhe.de, , , , , , DE

    G. Last

  2. Technische Universität Braunschweig, Institut für Mathematische Stochastik, 38023 Braunschweig, Germany, , , , , , DE

    R. Schassberger

Authors
  1. G. Last
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  2. R. Schassberger
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Received: 22 November 1999 / Revised version: 2 November 2000 /¶Published online: 14 June 2001

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Last, G., Schassberger, R. On the second derivative of the spherical contact distribution function of smooth grain models. Probab Theory Relat Fields 121, 49–72 (2001). https://doi.org/10.1007/PL00008797

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  • Issue Date: September 2001

  • DOI: https://doi.org/10.1007/PL00008797

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  • Mathematics Subject classification (2000): 60D05, 60G57
  • Key words or phrases: Spherical contact distribution – Grain model – Stochastic geometry – Palm probability – Surface measure – Random set – Random measure – Boolean model – Curvature measures
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