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Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $mathbb{S}^{d}subsetmathbb{R}^{d+1}$ 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 $mathbb{S}^{d}$.
Let ${B(xi_n,r_n)}_{nge1}$ be a sequence of random balls whose centers ${xi_n}_{nge1}$ is a stationary process, and ${r_n}_{nge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=limsuplimits_{ntoinfty}B(xi
We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radiu
Given $n$ points in the plane, a emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice even if th
Series expansions of isotropic Gaussian random fields on $mathbb{S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Lo`eve expansions of iso
We consider a variant of a classical coverage process, the boolean model in $mathbb{R}^d$. Previous efforts have focused on convergence of the unoccupied region containing the origin to a well studied limit $C$. We study the intersection of sets cent