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.