Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes

Abstract

Consider a random simplex [X_1,ldots ,X_n] defined as the convex hull of independent identically distributed (i.i.d.) random points X_1,ldots ,X_n in mathbb {R}^{n-1} with the following beta density: Let J_{n,k}(beta ) be the expected internal angle of the simplex [X_1,ldots ,X_n] at its face [X_1,ldots ,X_k]. Define {tilde{J}}_{n,k}(beta ) analogously for i.i.d. random points distributed according to the beta' density {tilde{f}}_{n-1,beta } (x) propto (1+Vert xVert ^2)^{-beta }, xin mathbb {R}^{n-1}, beta > ({n-1})/{2}. We derive formulae for J_{n,k}(beta ) and {tilde{J}}_{n,k}(beta ) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of beta . For J_{n,1}(pm 1/2) we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i) We compute explicitly the expected f-vectors of the typical Poisson–Voronoi cells in dimensions up to 10. (ii) Consider the random polytope K_{n,d} := [U_1,ldots ,U_n] where U_1,ldots ,U_n are i.i.d. random points sampled uniformly inside some d-dimensional convex body K with smooth boundary and unit volume. Reitzner (Adv. Math. 191(1), 178–208 (2005)) proved the existence of the limit of the normalised expected f-vector of K_{n,d}: lim _{nrightarrow infty } n^{-{({d-1})/({d+1})}}{mathbb {E}}{mathbf {f}}(K_{n,d}) = {mathbf {c}}_d cdot Omega (K), where Omega (K) is the affine surface area of K, and {mathbf {c}}_d is an unknown vector not depending on K. We compute {mathbf {c}}_d explicitly in dimensions up to d=10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.