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Distance measurements on processes of flats

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

Published online by Cambridge University Press:  01 July 2016

Daniel Hug ,
Günter Last  and
Wolfgang Weil
Daniel Hug*
Affiliation:
Albert-Ludwigs-Universität Freiburg
Günter Last*
Affiliation:
Universität Karlsruhe (TH)
Wolfgang Weil*
Affiliation:
Universität Karlsruhe (TH)
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, D-79104 Freiburg im Breisgau, Germany.
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany.
∗∗∗ Mathematisches Institut II, Universität Karlsruhe (TH), Englerstr. 2, D-76128 Karlsruhe, Germany. Email address: weil@math.uni-karlsruhe.de
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Abstract

Distance measurements are useful tools in stochastic geometry. For a Boolean model Z in ℝd, the classical contact distribution functions allow the estimation of important geometric parameters of Z. In two previous papers, several types of generalized contact distributions have been investigated and applied to stationary and nonstationary Boolean models. Here, we consider random sets Z which are generated as the union sets of Poisson processes X of k-flats, k ∈ {0, …, d-1}, and study distances from a fixed point or a fixed convex body to Z. In addition, we also consider the distances from a given flat or a flag consisting of flats to the individual members of X and investigate the associated process of nearest points in the flats of X. In particular, we discuss to which extent the directional distribution of X is determined by this point process. Some of our results are presented for more general stationary processes of flats.

Keywords

Stochastic geometrycontact distribution functionsupport (curvature) measurek-flatPoisson point processRadon transform

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Secondary: 60G57: Random measures 52A22: Random convex sets and integral geometry 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53C65: Integral geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 70 - 95
Copyright
Copyright © Applied Probability Trust 2003 

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