Limit shape in Poisson–Voronoi metrics over $\ell_p$ planes

Definitions and mathematical statement

Fix $p\in[1,\infty]$, and equip $\mathbb R^2$ with $\|x\|_p=(|x_1|^p+|x_2|^p)^{1/p}$ for finite $p$, or $\|x\|_\infty=\max(|x_1|,|x_2|)$. Let $\Pi$ be a homogeneous Poisson point process of intensity one with respect to planar Lebesgue area: counts in disjoint bounded sets are independent Poisson random variables with means equal to their areas.

For each nucleus $z\in\Pi$, form the closed Voronoi cell \[ C_z=\{x:\|x-z\|_p\le\|x-w\|_p\text{ for all }w\in\Pi\}. \] Two cells are adjacent when they share a boundary portion of positive length. Their adjacency graph has all edges of length one. Choose the cell containing the origin, using any deterministic tie convention on the null event of a tie, and let $\mathcal B_n$ be the union of cells at graph distance at most $n$ from it.

Determine the deterministic large-scale shape $K_p$ of these graph-metric balls. In particular, establish the appropriate shape theorem and identify $K_p$ so that \[ d_H(n^{-1}\mathcal B_n,K_p)\longrightarrow0 \quad\text{almost surely}, \] where $d_H$ is Hausdorff distance in the Euclidean plane. A description via directional time constants should determine their dependence on direction, not merely rename the unknown limit.

The adjacency metric counts cell crossings rather than Euclidean or $\ell_p$ length. Consequently the $\ell_p$ metric used to construct cells does not by itself determine the final ball shape. Even its scale is fixed here by Poisson intensity one and unit graph edges; an equivalent answer may describe the shape up to an explicitly stated normalization.

Short English statement

What macroscopic shape is produced by counting crossings of Poisson–Voronoi cells built using an arbitrary planar lp metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000057 (AMR-099-0057), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Benjamini - Euclidean vs. Graph Metric, Question 3.2, PDF page 4. Original source indexed by AMR-099-0057. https://www.wisdom.weizmann.ac.il/~itai/erd100.pdf#page=4.
• S3 I. Benjamini, Euclidean vs. Graph Metric, primary numbered question and conventions. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.