Abstract
Let X be the mosaic generated by a stationary Poisson hyperplane process X̂ in ℝd. Under some mild conditions on the spherical directional distribution of X̂ (which are satisfied if the process is isotropic), we show that with probability one the set of cells (d-polytopes) of X has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple d-polytopes is realized infinitely often by the cells of X. A further result concerns the distribution of the typical cell.
Communicated by: M. Henk
Acknowledgements
We thank the referee for useful hints, and Gilles Bonnet for helpful discussions. This work was partially supported by the DFG; it was initiated during the “Conference on Combinatorial Structures in Geometry” (Osnabrück) of the program DFG-RTG 1916.
References
[1] P. Erdös, A. Rényi, On Cantor’s series with convergent ∑1/qn. Ann. Univ. Sci. Budapest. Eötvös, Sect. Math. 2 (1959), 93–109. MR0126414 Zbl 0095.26501Search in Google Scholar
[2] A. Rényi, Wahrscheinlichkeitsrechnung. VEB Deutscher Verlag der Wissenschaften, Berlin 1977. MR0474442 Zbl 0396.60001Search in Google Scholar
[3] R. Schneider, W. Weil, Stochastic and integral geometry. Springer 2008. MR2455326 Zbl 1175.6000310.1007/978-3-540-78859-1Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
