Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-08T21:33:26.232Z Has data issue: false hasContentIssue false

Inequalities for mixed stationary Poisson hyperplane tessellations

Published online by Cambridge University Press:  01 July 2016

J. Mecke*
Affiliation:
University of Jena
*
Postal address: Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Institut für Stochastik, Ernst-Abbe Pl. 1-4, D-07743 Jena, Germany.

Abstract

Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 1998 

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). Line-processes, roads and fibres. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G.. Wiley, New York, pp. 248-251.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Mecke, J. (1986). On some inequalities for Poisson networks. Math. Nachr . 128, 8186.CrossRefGoogle Scholar
MECKE, J. (1995). Inequalities for the anisotropic Poisson polytope. Adv. Appl. Prob . 27, 5662.CrossRefGoogle Scholar
Mecke, J., Schneider, R., Stoyan, D., and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.CrossRefGoogle Scholar
MILES, R. E. (1969). Poisson flats in Euclidean spaces, Part I: A finite number of random uniform flats. Adv. Appl. Prob. 1, 211237.CrossRefGoogle Scholar
Miles, R. E. (1974). A synopsis of Poisson flats in Euclidean spaces. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G.. Wiley, New York, pp. 202-227.Google Scholar
SCHNEIDER, R. (1982). Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. 61, 379387.CrossRefGoogle Scholar
Stoyan, D., KENDALL, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications. Wiley, New York.Google Scholar
THOMAS, C. (1984). Extremum properties of the intersection densities of stationary Poisson hyperplane processes. Math. Operationforsch. Statist., Ser. Statist. 15, 443449.Google Scholar
VITALE, R. A. (1990). The Brunn-Minkowski inequality for random sets. J. Multivariate Anal. 33, 286293.CrossRefGoogle Scholar
WEIL, W. and Wieacker, J. A. (1993). Chapter 5.2 in Handbook of Convex Geometry, ed. Gruber, P. M. and Wills, J. M.. Elsevier, Amsterdam.Google Scholar