Spherical convex hull of random points on a wedge

TL;DR

This paper investigates the growth rate of the expected number of facets of the spherical convex hull formed by random points in a spherical wedge, showing it grows logarithmically with the number of points, similar to Euclidean and spherical cases.

Contribution

It establishes the asymptotic behavior of the expected facet number for random points in a spherical wedge, extending known results to this specific geometric setting.

Findings

Expected facet number grows like a constant times log n

Similar behavior observed for Poisson point processes

Comparison with Euclidean and spherical polytopes included

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.