Advances in Applied Probability

Approximation properties of random polytopes associated with Poisson hyperplane processes

Daniel Hug and Rolf Schneider

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 919-936.

Dates
First available in Project Euclid: 12 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1418396237

Digital Object Identifier
doi:10.1239/aap/1418396237

Mathematical Reviews number (MathSciNet)
MR3290423

Zentralblatt MATH identifier
1319.60015

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Poisson hyperplane process zero polytope approximation of convex bodies directional distribution

Citation

Hug, Daniel; Schneider, Rolf. Approximation properties of random polytopes associated with Poisson hyperplane processes. Adv. in Appl. Probab. 46 (2014), no. 4, 919--936. doi:10.1239/aap/1418396237. https://projecteuclid.org/euclid.aap/1418396237


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References

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