Ideal Poisson--Voronoi tessellations beyond hyperbolic spaces

TL;DR

This paper constructs and analyzes the properties of the ideal Poisson--Voronoi tessellation in the product space of two hyperbolic planes with an L^1 norm, revealing invariance and geometric features.

Contribution

It introduces the ideal Poisson--Voronoi tessellation in a non-hyperbolic setting and studies its invariance and geometric characteristics, extending previous results to this space.

Findings

The tessellation law is invariant under all isometries of the space.

The set of points equidistant to two corona points is almost surely unbounded.

The work extends known results to the product of hyperbolic planes with L^1 norm.

Abstract

We construct and study the ideal Poisson--Voronoi tessellation of the product of two hyperbolic planes $\mathbb{H}_{2}\times \mathbb{H}_{2}$ endowed with the $L^{1}$ norm. We prove that its law is invariant under all isometries of this space and study some geometric features of its cells. Among other things, we prove that the set of points at equal separation to any two corona points is unbounded almost surely. This is analogous to a recent result of Fr\k{a}czyk-Mellick-Wilkens for higher rank symmetric spaces.