Minimal dispersion on the sphere

TL;DR

This paper investigates the behavior of minimal spherical cap dispersion on the sphere as the number of points and dimension grow, connecting it to sphere covering and polytope approximation, with new bounds and insights.

Contribution

It introduces new bounds and methods for analyzing spherical cap dispersion, linking it to sphere covering and polytope approximation, and studies dispersion with cap intersections.

Findings

Upper bounds from random point selection and gap closing

Connections to sphere covering and polytope approximation

Analysis of dispersion with cap intersections

Abstract

The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with normalized area $\varepsilon$ not containing any of these points. We study the behavior of ${\rm disp}_{\mathcal{C}}(n,d)$ as $n$ and $d$ grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on ${\rm disp}_{\mathcal{C}}(n,d)$ result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.