Exact Scale--Shape Factorization of the Typical Poisson--Voronoi Cell Volume in Arbitrary Dimension

Abstract

Despite more than six decades of research, a tractable closed-form distribution for the typical Poisson--Voronoi cell volume remains unknown beyond one dimension. Building on the classical complementary-theorem structure for the Poisson--Voronoi fundamental region, we develop an explicit configuration-space factorization of the Palm-typical cell volume for a tessellation generated by a stationary Poisson point process of intensity \(λ>0\) in \(\mathbb R^d\), \(d\geq1\). Conditional on the number \(k\) of effective facets, the Voronoi flower volume is a \(\operatorname{Gamma}(k,\operatorname{rate}=λ)\) scale variable independent of the effective-neighbour configuration normalized to have unit flower volume. Mapping this normalized configuration to its cell-to-flower volume ratio \(A_k\) gives the conditional cell-volume representation as the product of the classical Gamma scale and a bounded geometric shape factor. From this representation, we derive exact mixture formulae, transform and moment identities, and criteria characterizing when the conditional cell-volume laws are Gamma. We also distinguish shape-factor variability within facet-number strata from mixing across strata as two sources of departure from a single Gamma law and recover the universal bound \(0<A_k\leq2^{-d}\). For the lower tail, we reduce critical negative moments of the shape factor to inverse-volume integrals over normalized configuration spaces. Under explicit critical-integrability and higher-facet summability assumptions, the density of the intensity-normalized cell volume has leading order \(y^d\) as \(y\downarrow0\). The framework recovers the one-dimensional distribution, admits explicit planar coordinates, and supports numerical evaluation of the mixture representation in dimensions two through four.

32 pages, 2 figures, 2 tables