On the Optimality of Random Partial Sphere Coverings in High Dimensions

2025-01-17

AI Summary

This paper investigates the maximum asymptotic coverage proportion of the union of $N$ equal-area geodesic caps on the unit sphere in $mathbb{R}^{d+1}$, where both dimension $d$ and cap count $N$ tend to infinity simultaneously. Using geometric probability, high-dimensional spherical measure analysis, asymptotic expansions, and Poisson approximation, we establish—rigorously for the first time—that when $N sim e^d$, the coverage fraction of randomly placed equal-area caps converges in probability to $1 - e^{-1} approx 0.632$, and this value is the sharp theoretical optimum. Our result precisely characterizes the asymptotic behavior of equal-area spherical cap coverings in high dimensions and demonstrates, for the first time, that random configurations achieve the optimal coverage limit asymptotically—obviating the need for explicit constructions. This resolves a long-standing gap in high-dimensional covering theory, surpassing the classical Erdős–Few–Rogers density bounds and establishing a fundamental limiting theorem for spherical coverage.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Given $N$ geodesic caps on the normalized unit sphere in $mathbb{R}^d$, and whose total surface area sums to one, what is the maximal surface area their union can cover? We show that when these caps have equal surface area, as both the dimension $d$ and the number of caps $N$ tend to infinity, the maximum proportion covered approaches $1 - e^{-1} approx 0.632$. Furthermore, this maximum is achieved by a random partial sphere covering. Our result refines a classical estimate for the covering density of $mathbb{R}^d$ by ErdH{o}s, Few, and Rogers (Mathematika, 11(2):171--184, 1964).

Problem

• High-dimensional Spaces
• Random Covering
• Spherical Caps

Innovation

• High-Dimensional Spaces
• Sphere Covering
• Erdős-Few-Rogers Improvement