Beta-star polytopes and hyperbolic stochastic geometry

Abstract

Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in R d with density proportional to ( x 2 − 1) −β , where x > 1 and β > d/2. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of k-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are relevant in the context of hyperbolic stochastic geometry, since they are tightly connected to the typical cell of a Poisson-Voronoi tessellation as well as the zero cell of a Poisson hyperplane tessellation in hyperbolic space. The general results for beta-star polytopes are used to provide explicit formulas for the expected fvector of the typical hyperbolic Poisson-Voronoi cell and the hyperbolic Poisson zero cell. Their asymptotics for large intensities and their monotonicity behaviour is discussed as well. Finally, stochastic geometry in the de Sitter half-space is studied as the hyperbolic analogue to recent investigations about random cones generated by random points on half-spheres in spherical or conical stochastic geometry.

Poisson-Voronoi percolation in the hyperbolic plane with small intensities

“…Works on continuum percolation models over Poisson processes in the hyperbolic plane include [5,16,42,44,45]. Aspects of hyperbolic Poisson-Voronoi tessellations besides percolation that have been studied include include the (expected) combinatorial structure of their cells, random walks on them and anchored expansions -see for example [6,7,13,19,26,27]. Percolation on hyperbolic Poisson-Voronoi tessellations was first studied specifically by Benjamini and Schramm in [9].…”

“…It might be possible to leverage some of the existing work on the expected f -vectors of the typical cell in [13,19,26]. But, of course it might also be possible to prove or disprove the conjecture without knowing the expected number of (d − 1)-faces precisely.…”