Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T06:28:06.060Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Daniel Hug  and
Rolf Schneider
Daniel Hug*
Affiliation:
Karlsruhe Institute of Technology
Rolf Schneider*
Affiliation:
Albert-Ludwigs-Universität Freiburg
*
∗ Postal address: Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany. Email address: daniel.hug@kit.edu
∗∗ Postal address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany. Email address: rolf.schneider@math.uni-freiburg.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

Keywords

Poisson hyperplane processzero polytopeapproximation of convex bodiesdirectional distribution

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 4 , December 2014 , pp. 919 - 936
Copyright
© Applied Probability Trust 

References

Bárány, I. (1989). Intrinsic volumes and f-vectors of random polytopes. Math. Ann. 285, 671–699.CrossRefGoogle Scholar
Böröczky, K. J. and Schneider, R. (2010). The mean width of circumscribed random polytopes. Canad. Math. Bull. 53, 614–628.Google Scholar
Dümbgen, L. and Walther, G. (1996). Rates of convergence for random approximations of convex sets. Adv. Appl. Prob. 28, 384–393.Google Scholar
Fáry, I. and Rédei, L. (1950). Der zentralsymmetrische Kern und die zentralsymmetrische Hülle von konvexen Körpern. Math. Ann. 122, 205–220.CrossRefGoogle Scholar
Hug, D. and Schneider, R. (2007). Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191.Google Scholar
Reitzner, M. (2010). Random polytopes. In New Perspectives in Stochastic Geometry, eds Kendall, W. S. and Molchanov, I., Oxford University Press, pp. 45–76,Google Scholar
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitsth. 2, 75–84.Google Scholar
Rényi, A. and Sulanke, R. (1964). Über die konvexe Hülle von n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitsth. 3, 138–147.CrossRefGoogle Scholar
Rényi, A. and Sulanke, R. (1968). Zufällige konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitsth. 9, 146–157.Google Scholar
Schneider, R. (2014). Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.Google Scholar
Werner, E. (1994). Illumination bodies and affine surface area. Studia Math. 110, 257–269.Google Scholar