Geometric properties of $N$ random points distributed independently anduniformly on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ withrespect to surface area measure are obtained and several related conjecturesare posed. In particular, we derive asymptotics (as $N \to \infty$) for theexpected moments of the radii of spherical caps associated with the facets ofthe convex hull of $N$ random points on $\mathbb{S}^{d}$. We provideconjectures for the asymptotic distribution of the scaled radii of thesespherical caps and the expected value of the largest of these radii (thecovering radius). Numerical evidence is included to support these conjectures.Furthermore, utilizing the extreme law for pairwise angles of Cai et al., wederive precise asymptotics for the expected separation of random points on$\mathbb{S}^{d}$.
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机译：获得关于单位面积独立且均匀分布在单位球面上的$ N $个随机点的几何性质，关于表面积测量，并得出了一些相关的猜想摆姿势。特别是，我们得出了与$ \ mathbb {S} ^ {d} $$上与$ N $凸点的凸包的小平面相关联的球形帽的半径的期望矩的渐近性（如$ N \ to \ infty $） 。我们提供了这些球形帽的缩放半径的渐近分布以及这些半径中的最大值的期望值（覆盖半径）的猜想。包括了数字证据来支持这些猜想。此外，利用Cai等人的成对角的极值定律，用精确的渐近渐进性来期望$ \ mathbb {S} ^ {d} $上随机点的分离。
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