Beta-star polytopes and hyperbolic stochastic geometry

TL;DR

This paper introduces beta-star polytopes, a new class of convex hulls in hyperbolic stochastic geometry, providing explicit formulas for their geometric functionals and connecting them to hyperbolic tessellations.

Contribution

The paper defines beta-star polytopes, derives explicit formulas for their geometric properties, and links them to hyperbolic Poisson-Voronoi and hyperplane tessellations.

Findings

Explicit formulas for face counts and angles of beta-star polytopes

Connections between beta-star polytopes and hyperbolic tessellation cells

Asymptotic behavior of geometric functionals for large intensities

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 $\mathbb{R}^d$ with density proportional to $(||x||^2-1)^\beta$, where $||x|| > 1$ and $\beta>d/2$. 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 $f$- vector 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.