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

Abstract

AbstractConsider a random simplex $$[X_1,\ldots ,X_n]$$ [ X 1 , … , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points $$X_1,\ldots ,X_n$$ X 1 , … , X n in $$\mathbb {R}^{n-1}$$ R n - 1 with the following beta density: "Equation missing" Let $$J_{n,k}(\beta )$$ J n , k ( β ) be the expected internal angle of the simplex $$[X_1,\ldots ,X_n]$$ [ X 1 , … , X n ] at its face $$[X_1,\ldots ,X_k]$$ [ X 1 , … , X k ] . Define $${\tilde{J}}_{n,k}(\beta )$$ J ~ n , k ( β ) analogously for i.i.d. random points distributed according to the beta$$'$$ ′ density $${\tilde{f}}_{n-1,\beta } (x) \propto (1+\Vert x\Vert ^2)^{-\beta }, x\in \mathbb {R}^{n-1}, \beta > ({n-1})/{2}.$$ f ~ n - 1 , β ( x ) ∝ ( 1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > ( n - 1 ) / 2 . We derive formulae for $$J_{n,k}(\beta )$$ J n , k ( β ) and $${\tilde{J}}_{n,k}(\beta )$$ J ~ n , k ( β ) which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of $$\beta $$ β . For $$J_{n,1}(\pm 1/2)$$ J n , 1 ( ± 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 $$K_{n,d} := [U_1,\ldots ,U_n]$$ K n , d : = [ U 1 , … , U n ] where $$U_1,\ldots ,U_n$$ U 1 , … , 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 $$K_{n,d}$$ K n , d : $$ \lim _{n\rightarrow \infty } n^{-{({d-1})/({d+1})}}{\mathbb {E}}{\mathbf {f}}(K_{n,d}) = {\mathbf {c}}_d \cdot \Omega (K),$$ lim n → ∞ n - ( d - 1 ) / ( d + 1 ) E f ( K n , d ) = c d · Ω ( K ) , where $$\Omega (K)$$ Ω ( K ) is the affine surface area of K, and $${\mathbf {c}}_d$$ c d is an unknown vector not depending on K. We compute $${\mathbf {c}}_d$$ c d explicitly in dimensions up to $$d=10$$ d = 10 and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.