Location of the most isolated point in a random polyhedral sample
Let $A \subset \mathbb{R}^3$ be a compact convex polyhedron with nonempty interior, and let $X_1, X_2, \dots$ be independent uniform points in $A$. Set $$R_{n,i} = \min_{j \ne i} \|X_i - X_j\|,$$ let $I_n$ be the smallest index attaining $\max_{1 \le i \le n} R_{n,i}$, and put $Z_n = X_{I_n}$, a most isolated sample point. For each edge $e$ of $A$ let $\alpha_e \in (0, \pi)$ be its interior dihedral angle, and set $$\alpha_* = \min_e \alpha_e, \qquad E_* = \bigcup_{\alpha_e = \alpha_*} e.$$ Penrose, Yang and Higgs proved that, almost surely, $$\frac{n \max_i R_{n,i}^3}{\log n} \longrightarrow |A| \max\left\{\frac{1}{\pi}, \frac{1}{2\alpha_*}\right\},$$ so that the sharpest edges determine the limit when $\alpha_* < \pi/2$ and the facets when $\alpha_* > \pi/2$.
Problem. Is it true that, if $\alpha_* > \pi/2$, then $Z_n$ converges in distribution to the normalized surface-area measure $\mathcal{H}^2|_{\partial A} / \mathcal{H}^2(\partial A)$, and that, if $\alpha_* < \pi/2$, then $Z_n$ converges in distribution to the normalized length measure $\mathcal{H}^1|_{E_*} / \mathcal{H}^1(E_*)$? Here $\mathcal{H}^s$ is $s$-dimensional Hausdorff measure. The critical case $\alpha_* = \pi/2$ is not part of the problem.
Source: Penrose, Yang and Higgs (2024), Introduction, p. 1724, stated as a belief without proof. Stated by AIM — Open Applied Problems with a smallest-index rule for ties, the critical angle π/2 excluded.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, convex polyhedra, nearest-neighbour distances, random geometric graphs
References
@article{penrose2024polytope,
author = {Penrose, Mathew D. and Yang, Xiaochuan and Higgs, Frankie},
title = {Largest nearest-neighbour link and connectivity threshold in a polytopal random sample},
journal = {J. Appl. Comput. Topol.},
volume = {8},
number = {6},
year = {2024},
pages = {1723--1750},
doi = {10.1007/s41468-023-00154-5},
eprint = {2301.02506},
archivePrefix = {arXiv},
}
@misc{aim-polyhedral-most-isolated-point-location,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Location of the most isolated point in a random polyhedral sample}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/648-polyhedral-most-isolated-point-location.md},
note = {Entry 648 at commit 24edaad, 2026-10-15},
}
Resolution claims
Let $A$ be a compact convex polyhedron in $\mathbb R^3$ with nonempty interior, and sample independent uniform points in $A$. We prove that the point with the largest nearest-neighbour distance converges in distribution to normalized surface area on $\partial A$ when every interior dihedral angle exceeds $\pi/2$. When the smallest angle is less than $\pi/2$, its limit is normalized length …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/polyhedral_isolation.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00819, "Location of the most isolated point in a random polyhedral sample", clopen.solutions.