Spherical convex hull of random points on a wedge

Abstract

Consider two half-spaces H1+ and H2+ in Rd+1 whose bounding hyperplanes H1 and H2 are orthogonal and pass through the origin. The intersection S2,+d:=Sd∩H1+∩H2+ is a spherical convex subset of the d-dimensional unit sphere Sd , which contains a great subsphere of dimension d- 2 and is called a spherical wedge. Choose n independent random points uniformly at random on S2,+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 S2,+d . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.