On the cells in a stationary Poisson hyperplane mosaic

Abstract

Abstract Let X be the mosaic generated by a stationary Poisson hyperplane process X̂ in ℝ d . 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.