Random Points in Halfspheres

Abstract

A random spherical polytope P-n in a spherically convex set K subset of 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. (C) 2016 Wiley Periodicals, Inc.