Typical Cells in Hyperbolic Tessellations with Unbounded Cells

Definitions and mathematical statement

Let $\mathbb H^d$ denote real hyperbolic space, with hyperbolic distance $d_{\mathbb H}$. The supplied record does not specify the permitted dimensions $d$. A nonempty cell $C\subseteq\mathbb H^d$ is unbounded if \[ \operatorname{diam}_{\mathbb H}(C):=\sup\{d_{\mathbb H}(x,y):x,y\in C\}=\infty. \] Let $T$ denote the hyperbolic tessellation considered in the source. Its precise admissible class and probabilistic assumptions are not provided in the supplied excerpt.

Construction problem. Find a natural extension of the notions of cell intensity and typical cell to hyperbolic tessellations $T$ containing unbounded cells. The extended notions should recover the existing notions on their original domain of definition. The intended target is a general extension of these notions, rather than a construction restricted to a particular example of a tessellation.

Unresolved qualifications. The supplied record does not define what makes an extension natural, state the original domain of definition, or specify the required invariance, local finiteness, convexity, integrability, or normalization assumptions. It also does not specify whether the typical cell is positioned, rooted, or considered up to isometry. Thus the construction problem cannot be sharpened into a fully quantified existence assertion from this record alone. The original excerpt contains a disrupted formula for the existing typical-cell distribution, referring to $\gamma_d(T)$, a cell center $z(C)$, a set $B$, and an isometry kernel $\kappa$; their definitions and the necessary hypotheses are missing, so that formula is not reconstructed here.

Short English statement

The problem asks for natural definitions of cell intensity and a typical cell when a hyperbolic tessellation has unbounded cells, compatible with the existing definitions where those apply. The supplied excerpt leaves the admissible tessellations and the meaning of naturality unspecified. This is a statement-only construction problem requiring mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30006510$, source problem code catalogue entry, title: Typical Cells in Hyperbolic Tessellations with Unbounded Cells. The supplied record labels the problem open and its statement corrected_verified; these are supplied curation labels, not independent verification. No individual proposer is identified.
• Supplied source citation: Mini-Workshop: Hyperbolic meets Stochastic Geometry (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/57, pp. 3031--3062. https://doi.org/10.4171/owr/2025/57. The original excerpt includes a page marker for p. 3047. The record gives proposed year 2025, while the citation title gives 2026; this discrepancy is unresolved.
• Supplied literature source: Oberwolfach Report on stochastic geometry. https://ems.press/content/serial-article-files/52444.
• Supplied literature source: Ideal Poisson--Voronoi tessellations in hyperbolic spaces. https://arxiv.org/abs/2303.16831.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and states that specific hyperbolic tessellations with unbounded cells are studied, but that no natural general extension of cell intensity and typical cell was established. This is supplied dated literature triage derived from the supplied OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.