Visibility and intersection density for Boolean models in hyperbolic space

TL;DR

This paper extends concepts like intersection density and visible volume from Euclidean to hyperbolic space for Poisson processes, providing conditions for finiteness based on intensity and grain surface area.

Contribution

It introduces hyperbolic space analogs of intersection density and visible volume and establishes a criterion for their finiteness in Boolean models.

Findings

Finiteness of mean visible volume characterized by intensity and surface area

Derived necessary and sufficient condition for Boolean models in hyperbolic space

Extended Euclidean concepts to hyperbolic geometric setting

Abstract

For Poisson particle processes in hyperbolic space we introduce and study concepts analogous to the intersection density and the mean visible volume, which were originally considered in the analysis of Boolean models in Euclidean space. In particular, we determine a necessary and sufficient condition for the finiteness of the mean visible volume of a Boolean model in terms of the intensity and the mean surface area of the typical grain.