Spherical convex hull of random points on a wedge

Abstract

Consider two half-spaces $H_1^{+}$ and $H_2^{+}$ in ${\mathbb{R}}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection ${\mathbb{S}}_{2,+}^d:={\mathbb{S}}^d\cap H_1^{+}\cap H_2^{+}$ is a spherical convex subset of the $d$-dimensional unit sphere ${\mathbb{S}}^d$, which contains a great subsphere of dimension $d-2$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on ${\mathbb{S}}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on ${\mathbb{S}}_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

21 pages, 5 figures