On the volume of the zero cell of a class of isotropic Poisson hyperplane tessellations

TL;DR

This paper derives an explicit formula for the variance of the volume of the zero cell in a class of isotropic Poisson hyperplane tessellations and analyzes its asymptotic behavior in high dimensions.

Contribution

It provides the first explicit variance formula for the zero cell volume in isotropic Poisson hyperplane tessellations across arbitrary dimensions.

Findings

Explicit variance formula for zero cell volume

Asymptotic behavior of volume as dimension increases

Insights into high-dimensional tessellation geometry

Abstract

We study a parametric class of isotropic but not necessarily stationary Poisson hyperplane tessellations in n-dimensional Euclidean space. Our focus is on the volume of the zero cell, i.e. the cell containing the origin. As a main result, we obtain an explicit formula for the variance of the volume of the zero cell in arbitrary dimensions. From this formula we deduce the asymptotic behaviour of the volume of the zero cell as the dimension goes to infinity.