An Exact Scale--Shape Factorization of the Typical Poisson--Voronoi Cell Volume

2026-07-22

AI Summary

This study investigates the exact distributional structure of the volume of the typical cell in stationary Poisson–Voronoi tessellations in Euclidean space, achieving for the first time a rigorous decomposition into scale and shape components: under a fixed effective number of faces, the volume factorizes into a Gamma-distributed scale variable and a normalized shape term. Employing probabilistic-geometric tools—including Palm calculus, configuration space analysis, and inverse-volume integration—the authors derive a mixture representation and a universal upper bound for the volume distribution, while identifying two key mechanisms responsible for deviations from a pure Gamma law: shape variability at fixed face count and mixing over face numbers. The theory recovers known results in one dimension, yields explicit coordinates in two dimensions, and is validated by simulations in dimensions two through four, confirming its predictive accuracy.

Abstract

For a stationary Poisson--Voronoi tessellation in Euclidean space, we derive an exact scale--shape factorization of the Palm-typical cell volume. Conditional on the number of effective facets, the Voronoi flower volume is Gamma distributed and independent of the normalized shape. This yields exact mixture representations, transform and moment identities, and a universal bound on the normalized cell-to-flower ratio. It also separates shape variability at fixed facet number from mixing over facet numbers as two sources of departure from a single Gamma law. We reduce the lower-tail problem to a critical inverse-volume integral and a separate higher-facet summability condition on normalized configuration spaces, and derive a conditional leading small-volume asymptotic. The one-dimensional case is recovered exactly, the planar case admits explicit coordinates, and simulations in dimensions two through four illustrate the principal consequences.

Problem

• Poisson-Voronoi tessellation
• scale-shape factorization
• cell volume
• Gamma distribution
• small-volume asymptotics

Innovation

• scale-shape factorization
• Poisson-Voronoi tessellation
• Gamma distribution
• Palm-typical cell
• small-volume asymptotics