Results about stationary Poisson hyperplane processes and the induced hyperplane mosaics are extended to the case where, instead of stationarity, it is only assumed that the intensity measure has a (possibly continuous) density with respect to some translation-invariant measure. Intensities and quermass densities, which are constant in the stationary case, are then replaced by functions. In a similar way, the associated zonoid (Matheron's Steiner convex set) is generalized.

Mixings of stationary Poisson hyperplane tessellations in d-dimensional Euclidean space are considered. The intention of the paper is to show that the 0-cell of a mixed stationary Poisson hyperplane tessellation Y is in some sense larger than that of stationary Poisson hyperplane tessellations Y' with the same intensity and directional distribution as Y. Related results concerning the moments for the volume of the 0-cell are derived. In special cases, similar statements with respect to the typical cell are proved.

The typical cell of a stationary Poisson hyperplane tessellation in the d-dimensional Euclidean space is called the Poisson polytope, and the cell containing the origin is called the Poisson 0-polytope. The intention of the paper is to show that the cells of the anisotropic tessellations are in some sense larger than those of the isotropic tessellations. Under the condition of equal intensities, it is proved that the moments of order n = 1, 2, … for the volume of the Poisson 0-polytope in the anisotropic case are not smaller than the corresponding moments in the isotropic case. Similar results are derived for the Poisson polytope. Finally, generalizations are mentioned.