Face Numbers and Volumes of Poisson Zero Cells

Definitions and mathematical statement

For a tessellation of ${\mathbb{R}}^d$, its zero cell is the cell containing the origin, unique almost surely in the usual stationary models. It is not the Palm typical cell. For a bounded $d$-polytope $P$, let $f_\ell(P)$ denote the number of its $\ell$-dimensional faces. The source poses two separate moment questions.

1. For a Poisson hyperplane tessellation, write $Z_0^{\mathrm H}$ for its zero cell. In the standard construction it is the intersection of the halfspaces containing the origin bounded by the process hyperplanes. Determine ${\mathbb{E}} f_\ell(Z_0^{\mathrm H})$ for $\ell\geq1$. The nontrivial proper-face range is $1\leq\ell\leq d-1$; if the whole polytope is counted as a face, $f_d=1$, and $f_\ell=0$ for $\ell>d$. The dimension, hyperplane intensity/directional distribution, and nondegeneracy assumptions making the cell bounded are not given. No isotropic specialization is silently imposed.
2. For a Poisson--Voronoi mosaic, write $Z_0^{\mathrm V}$ for its zero cell and determine its mean volume and face numbers. In the usual Euclidean model generated by a homogeneous Poisson point process $\xi$ of intensity $\lambda>0$, cells are \[C(x)=\{y\in{\mathbb{R}}^d:|y-x|\leq|y-z|\text{ for every }z\in\xi\},\qquad x\in\xi,\] and $Z_0^{\mathrm V}$ is the cell containing the origin. The requested quantities are ${\mathbb{E}}\operatorname{Vol}_d(Z_0^{\mathrm V})$ and ${\mathbb{E}} f_\ell(Z_0^{\mathrm V})$ in the intended face range. This sampling differs from adjoining a nucleus at the origin and taking its cell; typical-cell moments are not substituted. The source does not state the dimension or intensity normalization or explicitly specify stationarity.

The questions seek exact mean quantities for the intended models, not only large-cell asymptotics. The supplied 2026-08-21 triage labels the compound record partially solved, reports a proved large Poisson--Voronoi circle-shape conjecture and later large-zero-cell asymptotics, and says the extracted passage also recorded missing exact zero-cell moments. That separate shape/asymptotic progress is preserved as a qualification, not treated as an answer to every moment request or as a new displayed conjecture.

Short English statement

Determine mean nonvertex face numbers for Poisson-hyperplane zero cells and mean volume and face numbers for Poisson--Voronoi zero cells. The dimension and process parameters are unstated, and zero-cell sampling must not be replaced by typical-cell sampling.

Sources

• Local UnsolvedMath registry, permanent ID 30001083, source problem catalogue entry, Face Numbers and Volumes of Poisson Zero Cells, proposed year 2008; no individual proposer supplied.
• Supplied citation: New Perspectives in Stochastic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/47, pp. 2655--2702: https://doi.org/10.4171/owr/2008/47. Supplied literature label: original OWR report.
• Supplied literature label: Hug--Reitzner--Schneider, large Poisson--Voronoi cells, https://doi.org/10.1239/aap/1093962228.
• Supplied literature label: Calka--Demichel--Enriquez, large planar zero cells, https://www.numdam.org/item/10.5802/ahl.86.pdf.
• Literature assessment date: 2026-08-21. The status and progress claims are supplied dated curation, not independently verified here.