Angles of random simplices and face numbers of random polytopes

Pick d+1 points uniformly at random on the unit sphere in Rd. What is expected value of angle sum simplex spanned by these points? Choose n d-dimensional ball. number faces their convex hull? We answer and some other, seemingly unrelated, questions stochastic geometry. To this end, we compute internal angles simplices whose vertices are independent sampled from one following distributions: (i) beta distribution with density proportional to (1−‖x‖2)β, where x belongs ball Rd; (ii) beta' (1+‖x‖2)−β whole These results imply explicit formulae for face numbers polytopes: (a) typical Poisson-Voronoi cell; (b) zero cell Poisson hyperplane tessellation; (c) polytopes defined as hulls i.i.d. samples corresponding distributions.