Poisson-Laguerre tessellations

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 $f$.