Spherical convex hull of random points on a wedge

Consider two half-spaces H 1 + and H 2 + in R d + 1 whose bounding hyperplanes H 1 and H 2 are orthogonal and pass through the origin. The intersection S 2 , + d : = S d ∩ H 1 + ∩ H 2 + is a spherical convex subset of the d -dimensional unit sphere 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 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 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.