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

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.