2025/09/23 by Mathew D. Penrose, Penrose, Mathew D., Xiaochuan Yang +1 · 1 citation
Computer Science · Mathematics · #53A05 #60D05 #60F05 #60F15 #60G55 #Computational Geometry and Mesh Generation #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2509.19278
openalex publication_date 2025/09/23 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
Let A be a compact d-dimensional C2 Riemannian manifold with boundary, embedded in \bf Rm where m ≥ d ≥ 2, and let B be a nice subset of A (possibly B=A). Let X1,X2, … be independent random uniform points in A. Define the coverage threshold Rn to be the smallest r such that B is covered by the geodetic balls of radius r centred on X1,…,Xn. We obtain the limiting distribution of Rn and also a strong law of large numbers for Rn in the large-n limit. For example, if A has Riemannian volume 1 and its boundary has surface measure |∂ A|, and B=A, then if d=3 then \bf P[nπRn3 - log n - 2 log (log n) ≤ x] converges to exp(-2-4π5/3 |∂ A| e-2 x/3) and (n πRn3)/(log n) → 1 almost surely, while if d=2 then \bf P[n πRn2 - log n - log (log n) ≤ x] converges to exp(- e-x- |∂ A|π-1/2 e-x/2). We generalize to allow for multiple coverage. For the strong laws of large numbers, we can relax the requirement that the underlying density on A be uniform. For the limiting distribution, we have a similar result for Poisson samples. Our results still hold if we use Euclidean rather than geodetic balls.