Angles of Random Simplices and Face Numbers of Random Polytopes

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