Minimal dispersion on the sphere

Abstract

The minimal spherical cap dispersion ${\mathrm{disp}}_{\mathcal{C}}(n,d)$ is the largest number $\epsilon \in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere ${\mathbb{S}}^d$, there exists a spherical cap with normalized area $\epsilon$ not containing any of these points. We study the behavior of ${\mathrm{disp}}_{\mathcal{C}}(n,d)$ 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 ${\mathrm{disp}}_{\mathcal{C}}(n,d)$ 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.

30 pages