Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T13:20:28.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonstationary Poisson hyperplanes and their induced tessellations

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, Germany. Email address: rolf.schneider@math.uni-freiburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Results about stationary Poisson hyperplane processes and the induced hyperplane mosaics are extended to the case where, instead of stationarity, it is only assumed that the intensity measure has a (possibly continuous) density with respect to some translation-invariant measure. Intensities and quermass densities, which are constant in the stationary case, are then replaced by functions. In a similar way, the associated zonoid (Matheron's Steiner convex set) is generalized.

Keywords

Hyperplane processPoisson processintensityintersection processhyperplane mosaicface processquermass densityassociated zonoidSteiner convex set

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 139 - 158
Copyright
Copyright © Applied Probability Trust 2003 

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

Capasso, V. and Micheletti, A. (2000). The local mean volume and surface densities for inhomogeneous random sets. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 65, 49–66.Google Scholar
Capasso, V. and Micheletti, A. (2000). Local spherical contact distribution function and local mean densities for inhomogeneous random sets. Stoch. Stoch. Rep. 71, 51–67.CrossRefGoogle Scholar
Cramér, H. and Wold, H. (1936). Some theorems on distribution functions. J. London Math. Soc. 11, 290–294.CrossRefGoogle Scholar
Fallert, H. (1992). Intensitätsmaße und Quermaßdichten für (nichtstationäre) zufällige Mengen und geometrische Punktprozesse. Dissertation, Universität Karls-ruhe.Google Scholar
Fallert, H. (1996). Quermaßdichten für Punktprozesse konvexer Körper und Boolesche Modelle. Math. Nachr. 181, 165–184.Google Scholar
Favis, W. (1995). Extremaleigenschaften und Momente für stationäre Poissonsche Hyperebenenmosaike. Dissertation, Friedrich-Schiller-Universität Jena.Google Scholar
Favis, W. and Weiss, V. (1998). Mean values of weighted cells of stationary Poisson hyperplane tessellations of Rd . Math. Nachr. 93, 37–48.Google Scholar
Goodey, P. and Howard, R. (1990). Processes of flats induced by higher dimensional processes. Adv. Math. 80, 92–109.Google Scholar
Hahn, U. and Stoyan, D. (1998). Unbiased stereological estimation of the surface area of gradient surface processes. Adv. Appl. Prob. 30, 904–920.Google Scholar
Hahn, U., Micheletti, A., Pohlink, R., Stoyan, D. and Wendrock, H. (1999). Stereological analysis and modelling of gradient structures. J. Microscopy 195, 113–124.Google Scholar
Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Prob. 28, 796–850.Google Scholar
Hug, D., Last, G. and Weil, W. (2002). Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Prob. 34, 21–47.Google Scholar
Keutel, J. (1991). Ein Extremalproblem für zufällige Ebenen und für Ebenenprozesse in höherdimensionalen Räumen. Dissertation, Friedrich-Schiller-Universität Jena.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mecke, J. (1983). Inequalities for intersection densities of superpositions of stationary Poisson hyperplane processes. In Proc. 2nd Intnat. Work-shop Stereology, Stochastic Geometry (Aarhus, October 1983), eds Jensen, E. B. and Gundersen, H. J. G., Institute of Mathematics Aarhus University, pp. 115–124.Google Scholar
Mecke, J. (1984). Random tessellations generated by hyperplanes. In Stochastic Geometry, Geometric Statistics, Stereology (Oberwolfach, 1983), eds Ambartzumian, R. V. and Weil, W., Teubner, Leipzig, pp. 104–109.Google Scholar
Mecke, J. (1986). On some inequalities for Poisson networks. Math. Nachr. 128, 81–86.Google Scholar
Mecke, J. (1988). An extremal property of random flats. J. Microscopy 151, 205–209.CrossRefGoogle Scholar
Mecke, J. (1988). Random r-flats meeting a ball. Arch. Math. 51, 378–384.Google Scholar
Mecke, J. (1991). On the intersection density of flat processes. Math. Nachr. 151, 69–74.Google Scholar
Mecke, J. (1995). Inequalities for the anisotropic Poisson polytope. Adv. Appl. Prob. 27, 56–62.Google Scholar
Mecke, J. (1998). Inequalities for mixed stationary Poisson hyperplane tessellations. Adv. Appl. Prob. 30, 921–928.Google Scholar
Mecke, J. (1999). On the relationship between the 0-cell and the typical cell of a stationary random tessellation. Pattern Recognition 32, 1645–1648.Google Scholar
Mecke, J. and Nagel, W. (1984). Reconstruction of planar line measures from section measures I. Res. Rep. N/84/10, Friedrich-Schiller-Universität Jena.Google Scholar
Mecke, J. and Thomas, C. (1986). On an extreme value problem for flat processes. Commun. Statist. Stoch. Models 2, 273–280.CrossRefGoogle Scholar
Micheletti, A. and Stoyan, D. (1998). Volume fraction and surface density for inhomogeneous random sets. Quaderno N. 17, Dipartimento di Matematica ‘F. Enriques’, Università degli Studi di Milano.Google Scholar
Miles, R. E. (1970). A synopsis of ‘Poisson flats in Euclidean spaces’. Izv. Akad. Nauk Arm. SSR, Ser. Mat. 5, 263–285.Google Scholar
Miles, R. E. (1971). Poisson flats in Euclidean spaces. II: Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 3, 1–43.CrossRefGoogle Scholar
Nagel, W. (1983). On the stereology for the distribution of line processes in the plane. Res. Rep. N/83/13, Friedrich-Schiller-Universität Jena.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.Google Scholar
Schneider, R. (2001). Crofton formulas in hypermetric projective Finsler spaces. Arch. Math. 77, 85–97.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Schwella, A. (2001). Special inequalities for Poisson and Cox hyperplane processes. Dissertation, Friedrich-Schiller-Universität Jena.Google Scholar
Spodarev, E. (2001). Selected topics in the theory of spatial stationary flat processes. Dissertation, Friedrich-Schiller-Universität Jena.Google Scholar
Spodarev, E. (2001). On the rose of intersections of stationary flat processes. Adv. Appl. Prob. 33, 584–599.Google Scholar
Thomas, C. (1983). Stationäre Hyperebenenprozesse im Rn—Extremaleigenschaften, Überlagerungen und stereologische Probleme. Dissertation, Friedrich-Schiller-Universität Jena.Google Scholar
Thomas, C. (1984). Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operationsforsch. Statist., Ser. Statist. 15, 443–449.Google Scholar
Weil, W. (1987). Point processes of cylinders, particles and flats. Acta Appl. Math. 9, 103–136.Google Scholar
Weil, W. (2000). A uniqueness problem for non-stationary Boolean models. Rend. Circ. Mat. Palermo, Ser. II, Suppl. 65, 329–344.Google Scholar
Weil, W. (2000). Mixed measures and inhomogeneous Boolean models. In Statistical Physics and Spatial Statistics, eds Mecke, K. R. and Stoyan, D., Springer, Berlin, pp. 95–110.CrossRefGoogle Scholar
Weil, W. (2001). Densities of mixed volumes for Boolean models. Adv. Appl. Prob. 33, 39–60.Google Scholar
Weiss, V. (1986). Relations between mean values for stationary random hyperplane mosaics of Rd . Res. Rep. N/86/33, Friedrich-Schiller-Universität Jena.Google Scholar
Weiss, V. and Zähle, M. (1988). Geometric measures for random curved mosaics of Rd . Math. Nachr. 138, 313–326.CrossRefGoogle Scholar
Zaanen, A. C. (1967). Integration. North-Holland, Amsterdam.Google Scholar