Minimal Dispersion on the Sphere.

Abstract

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