Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T02:56:37.771Z Has data issue: false hasContentIssue false
  • English
  • Français

A Duality for Poisson Flats

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Rolf Schneider
Rolf Schneider*
Affiliation:
Albert-Ludwigs Universität, Freiburg
*
∗ Postal address: Mathematisches Institut, Albert-Ludwigs Universität, Eckerstr. 1, D 79104 Freiburg i Br, Germany. Email address: rschnei@ruf.uni-freiburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In keeping with the intersection density of a stationary Poisson process of r-flats in Euclidean d-space, where r ≥ d/2, we introduce a notion of closeness, called proximity, for such processes if r < d/2. It is shown that the two notions are connected by a duality: the proximity of a stationary Poisson r-flat process is, up to a constant factor, the intersection density of a certain unique stationary Poisson (d − r)-flat process.

Keywords

Poisson flat processintersection density

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 63 - 68
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Davidson, R. (1974). Stochastic processes of flats and exchangeability. In Stochastic Geometry, eds. Harding, E. F. and Kendall, D. G. Wiley, London, pp. 13–45.Google Scholar
Davidson, R. (1974). Line-processes, roads and fibres. In Stochastic Geometry, eds. Harding, E. F. and Kendall, D. G. Wiley, London, pp. 248–251.Google Scholar
Janson, S. and Kallenberg, O. (1981). Maximizing the intersection density of fibre processes. J. Appl. Prob. 18, 820–828.CrossRefGoogle Scholar
Keutel, J. (1991). Ein Extremalproblem für zufällige Ebenen und für Ebenenprozesse in höherdimensionalen Räumen. Ph.D. Dissertation, Universität Jena.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Mecke, J. (1988). Random r-flats meeting a ball. Arch. Math. 51, 378–384.Google Scholar
Mecke, J. (1988). An extremal property of random flats. J. Microscopy 151, 205–209.CrossRefGoogle Scholar
Mecke, J. (1991). On the intersection density of flat processes. Math. Nachr. 151, 69–74.Google Scholar
Mecke, J. and Thomas, C. (1986). On an extreme value problem for flat processes. Commun. Statist. Stochastic Models 2, 273–280.CrossRefGoogle Scholar
Miles, R. (1964). Random polygons determined by random lines in a plane. Proc. Nat. Acad. Sci. USA 52, 1157–1160.CrossRefGoogle ScholarPubMed
Thomas, C. (1984). Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operationsforsch. Statist., Ser. Statist. 15, 443–449.Google Scholar