Spherical convex hull of random points on a wedge

Abstract Consider two half-spaces $$H_1^+$$ H 1 + and $$H_2^+$$ 2 in $${\mathbb {R}}^{d+1}$$ R d whose bounding hyperplanes $$H_1$$ $$H_2$$ are orthogonal pass through the origin. The intersection {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ S , : = ∩ is a spherical convex subset of d -dimensional unit sphere {S}}^d$$ , which contains great subsphere dimension $$d-2$$ - called wedge. Choose n independent random points uniformly at on {S}}_{2,+}^d$$ consider expected facet number hull these points. It shown that, up to terms lower order, this expectation grows like constant multiple $$\log n$$ log n . A similar behaviour obtained for homogeneous Poisson point process result compared corresponding classical Euclidean polytopes half-sphere.