Random points in halfspheres

TL;DR

This paper studies the asymptotic properties of random spherical polytopes formed by convex hulls of points in halfspheres, highlighting differences from Euclidean cases and providing estimates for various geometric characteristics.

Contribution

It establishes the asymptotic behaviour of key geometric features of random spherical polytopes in halfspheres, a case distinct from Euclidean analogs.

Findings

Asymptotic expectation of facet and vertex counts

Volume and surface area estimates for large n

Almost sure asymptotic estimates for Hausdorff distance

Abstract

A random spherical polytope $P_n$ in a spherically convex set $K \subset S^d$ as considered here is the spherical convex hull of $n$ independent, uniformly distributed random points in $K$. The behaviour of $P_n$ for a spherically convex set $K$ contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when $K$ is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as $n$ tends to infinity, of the expectation of several characteristics of $P_n$, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates.