Special inequalities for Poisson and Cox hyperplane processes (Q2784253)

We consider secondary quantities of stationary Poisson hyperplane processes (SPHP) and its induced tessellations such as intersection densities, the 0-polytope and the typical polytope. In the first part, we focus on the consideration of the moments of the volume of the 0-polytope and the typical polytope. We estimate these moments for general SPHP and give even more restrictive upper bounds for 3-dimensional SPHP which consist of planes parallel to the faces of a cube, a tetrahedron, a dodecahedron or an icosahedron, respectively. Furthermore, a proof is provided for the fact that the sequence of the moments determines the distribution of the volume of the 0-polytope uniquely in the case of a 2-dimensional SPHP. The second part is devoted to another type of secondary quantities, the intersection densities \(\rho_k\). For a \(d\)-dimensional hyperplane process, a secondary network of \((d-k)\)-dimensional planes is formed by the set of all intersections of any \(k\) hyperplanes in a general position. Heuristically, the quantity \(\rho_k\) reflects the mean \((d-k)\)-content of these intersections per unit volume.NEWLINENEWLINENEWLINEDue to the beautiful connection between the distribution of a SPHP and a particular convex body, the so-called Steiner compact set, it is possible to translate questions concerning intersection densities into questions concerning convex bodies. The intersection density \(\rho_k\) corresponds to the intrinsic \(k\)-volume \(V_k\) or, more common, to the \((d-k)\)th Minkowski functional, of the Steiner compact set. Furthermore, superposition of hyperplane processes corresponds to Minkowski addition of convex sets. Therefore, the famous Brunn-Minkowski inequality and its linear improvements play a central role in this work. Also, conditions for convex bodies \(K,L\) under which NEWLINE\[NEWLINE{V_k\over V_{k-1}} (K+L)\geq {V_k\over V_{k-1}}(K)+ {V_k \over V_{k-1}}(L)NEWLINE\]NEWLINE holds true are derived. These geometric facts are translated into the language of hyperplane processes. The last part is devoted to a generalization of the concept of SPHP, that is, to special Cox processes, or mixed SPHP. It turns out that various extremum properties of SPHP still hold in this much more general setting. Furthermore, new secondary quantities are introduced for these hyperplane processes, and their properties are investigated.