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A connection between the volume fractions of the Stienen model and the dead leaves model

Part of: Stochastic processes Geometric probability and stochastic geometry General convexity

Published online by Cambridge University Press:  01 July 2016

Marianne Månsson
Marianne Månsson*
Affiliation:
Chalmers University of Technology and Göteborg University
*
∗ Current address: Sagmastaregaten 1 i, SE 41680 Göteborg, Sweden. Email address: marianne@math.chalmers.se
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Abstract

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The volume fraction of the intact grains of the dead leaves model with spherical grains of equal size is 2−d in d dimensions. This is the volume fraction of the original Stienen model. Here we consider some variants of these models: the dead leaves model with grains of a fixed convex shape and possibly random sizes and random orientations, and a generalisation of the Stienen model with convex grains growing at random speeds. The main result of this paper is that if the radius distribution in the dead leaves model equals the speed distribution in the Stienen model, then the volume fractions of the two models are the same in this case also. Furthermore, we show that for grains of a fixed shape and orientation, centrally symmetric sets give the highest volume fraction, while simplices give the lowest. If the grains are randomly rotated, then the volume fraction achieves its highest value only for spheres.

Keywords

Dead leaves modelStienen modelMatérn hard core processnonintersecting grainsvolume fractionconvex grain

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 52A22: Random convex sets and integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 41 - 52
Copyright
Copyright © Applied Probability Trust 2007 

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