Accessible Unlicensed Requires Authentication Published by De Gruyter July 20, 2018

On the cells in a stationary Poisson hyperplane mosaic

Matthias Reitzner and Rolf Schneider
From the journal Advances in Geometry

Abstract

Let X be the mosaic generated by a stationary Poisson hyperplane process in ℝd. Under some mild conditions on the spherical directional distribution of (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.

  1. 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 0095.26501 Search in Google Scholar

[2] A. Rényi, Wahrscheinlichkeitsrechnung. VEB Deutscher Verlag der Wissenschaften, Berlin 1977. MR0474442 Zbl 0396.60001 Search in Google Scholar

[3] R. Schneider, W. Weil, Stochastic and integral geometry. Springer 2008. MR2455326 Zbl 1175.60003 Search in Google Scholar

Received: 2016-09-14
Revised: 2017-02-27
Published Online: 2018-07-20
Published in Print: 2019-04-24

© 2019 Walter de Gruyter GmbH, Berlin/Boston