Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation

Abstract

Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as $N \to \infty$) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of $N$ random points on $\mathbb{S}^{d}$. We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on $\mathbb{S}^{d}$.