Poisson-Laguerre tessellations

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 $(v,h)\mapsto f(h)\mathrm{d}h\mathrm{d}v$, 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 $f$.