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