Advances in Applied Probability

Faces with given directions in anisotropic Poisson hyperplane mosaics

Daniel Hug and Rolf Schneider

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, ..., d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k < d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

Article information

Adv. in Appl. Probab. Volume 43, Number 2 (2011), 308-321.

First available in Project Euclid: 21 June 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Poisson hyperplane tessellation volume-weighted typical face typical face conditional distribution Blaschke body shape


Hug, Daniel; Schneider, Rolf. Faces with given directions in anisotropic Poisson hyperplane mosaics. Adv. in Appl. Probab. 43 (2011), no. 2, 308--321.

Export citation


  • Baumstark, V. and Last, G. (2007). Some distributional results for Poisson–Voronoi tessellations. Adv. Appl. Prob. 39, 16–40.
  • Hug, D. and Schneider, R. (2007). Asymptotic shapes of large cells in random tessellations. Geom. Funct. Anal. 17, 156–191.
  • Hug, D. and Schneider, R. (2007). Typical cells in Poisson hyperplane tessellations. Discrete Comput. Geom. 38, 305–319.
  • Hug, D. and Schneider, R. (2010). Large faces in Poisson hyperplane mosaics. Ann. Prob. 38, 1320–1344.
  • Hug, D., Reitzner, M. and Schneider, R. (2004). The limit shape of the zero cell in a stationary Poisson hyperplane tessellation. Ann. Prob. 32, 1140–1167.
  • Hug, D., Reitzner, M. and Schneider, R. (2004). Large Poisson–Voronoi cells and Crofton cells. Adv. Appl. Prob. 36, 667–690.
  • Kovalenko, I. N. (1997). Proof of David Kendall's conjecture concerning the shape of large random polygons. Cybernet. Systems Anal. 33, 461–467.
  • Miles, R. E. (1964). Random polygons determined by random lines in a plane. II. Proc. Nat. Acad. Sci. USA 52, 1157–1160.
  • Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • Møller, J. (1989). Random tessellations in $\mathbb R^d$. Adv. Appl. Prob. 21, 37–73.
  • Schneider, R. (1993). Convex Bodies: the Brunn–Minkowski Theory. Cambridge University Press.
  • Schneider, R. (2009). Weighted faces of Poisson hyperplane tessellations. Adv. Appl. Prob. 41, 682–694.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.