Poisson-Laguerre tessellations

TL;DR

This paper introduces Poisson-Laguerre tessellations generated by a Poisson process in Euclidean space with a specific intensity measure, analyzing their sectional properties and deriving explicit distribution formulas for typical cells.

Contribution

It presents a new family of Poisson-Laguerre tessellations, explores their sectional properties, and provides explicit distribution representations for their typical cells.

Findings

Sectional properties relate to fractional integrals of the intensity function.

Explicit distribution formulas involve fractional calculus of the intensity function.

The framework generalizes classical tessellations with new probabilistic and geometric insights.

Abstract

In this paper we introduce a family of Poisson-Laguerre tessellations in $\mathbb{R}^d$ generated by a Poisson point process in $\mathbb{R}^d\times \mathbb{R}$, whose intensity measure has a density of the form $(v,h)\mapsto f(h){\rm d} h {\rm d} v$, where $v\in\mathbb{R}^d$ and $h\in\mathbb{R}$, with respect to the Lebesgue measure. We study its sectional properties and show that the $\ell$-dimensional section of a Poisson-Laguerre tessellation corresponding to $f$ is an $\ell$-dimensional Poisson-Laguerre tessellation corresponding to $f_{\ell}$, which is up to a constant a fractional integral of $f$ of order $(d-\ell)/2$. Further we derive an explicit representation for the distribution of the volume weighted typical cell of the dual Poisson-Laguerre tessellation in terms of fractional integrals and derivatives of $f$.