Poisson-Laguerre tessellations

In this paper we introduce a family of Poisson-Laguerre tessellations in Rd generated by a Poisson point process in Rd×R, whose intensity measure has a density of the form (v,h)↦f(h)dhdv, where v∈Rd and h∈R, with respect to the Lebesgue measure. We study its sectional properties and show that the ℓ-dimensional section of a Poisson-Laguerre tessellation corresponding to f is an ℓ-dimensional Poisson-Laguerre tessellation corresponding to fℓ, which is up to a constant a fractional integral of f of order (d−ℓ)∕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.