Poisson-Laguerre tessellations
Anna Gusakova, Mathias in Wolde-Lübke
TL;DR
The paper extends stochastic tessellation theory by introducing Poisson-Laguerre tessellations generated from a Poisson point process on $\mathbb{R}^d\times\mathbb{R}$ with a height-dependent density $f(h)$. It establishes that affine sections preserve the Laguerre structure via fractional integrals and provides an explicit sectional correspondence $\mathcal{L}_{d,\gamma}(f)\cap L \overset{d}{=}\mathcal{L}_{\ell,\gamma}(f_{\ell})$ with $f_{\ell}$ given by a fractional integral of order $(d-\ell)/2$, enabling dimension-reduced analysis. For the dual model, it derives a stochastic representation for the volume-weighted typical cell $Z_{d,\gamma,\nu}(f)$ in terms of fractional integrals/derivatives of $f$, and offers a canonical decomposition that recovers the classical $\beta$-, $\beta'$-, and Gaussian Laguerre/Delaunay tessellations. These results unify and generalize prior work (e.g., Gaussian, $\beta$, and $\beta'$ models) and provide tractable formulas for moments and distributions of important geometric characteristics, with potential applications in materials science, imaging, and stochastic geometry.
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 $f$.