On the Optimality of Random Partial Sphere Coverings in High Dimensions

TL;DR

This paper establishes an asymptotically sharp upper bound on the coverage of the high-dimensional sphere by random caps, revealing that nearly 63.2% of the sphere can be covered with high probability as dimension grows.

Contribution

It provides the first asymptotically sharp upper bound for partial sphere coverings in high dimensions, connecting geometric probability with theoretical computer science.

Findings

Maximum coverage approaches 1 - e^{-1} as dimension increases.

The result relates to the optimality of random polytopes in high dimensions.

Limitations are discussed via Gaussian surface area bounds.

Abstract

Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, and whose total normalized surface area sums to one, what is the maximal surface area their union can cover? In this work, we provide an asymptotically sharp upper bound for an antipodal partial covering of the sphere by $N \in (\omega(1),e^{o(\sqrt{d})})$ congruent caps, showing that the maximum proportion covered approaches $1 - e^{-1}$ as $d\to\infty$. We discuss the relation of this result to the optimality of random polytopes in high dimensions, the limitations of our technique via the Gaussian surface area bounds of K. Ball and F. Nazarov, and its applications in computer science theory.