Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Abstract

Let $$U_1,U_2,\ldots $$ be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $$n\rightarrow \infty $$ , the f-vector of the $$(d+1)$$ -dimensional convex cone $$C_n$$ generated by $$U_1,\ldots ,U_n$$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of $$C_n$$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $$C_n$$ can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644 ). Our approach is based on the observation that the random cone $$C_n$$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $$\Vert x\Vert ^{-(d+\gamma )}$$ , where $$\gamma =1$$ . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary $$\gamma >0$$ .