Let U 1 , U 2 , . . . be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as n → ∞, the f -vector of the (d + 1)-dimensional convex cone C n generated by U 1 , . . . , 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 Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. 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 x −(d+γ) , where γ = 1. We compute the expected number of facets, the expected intrinsic volumes and the expected T -functional of this random convex hull for arbitrary γ > 0.

Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,βfalse(xfalse)=const·()1−∥x∥2β,false∥xfalse∥<1,(thebetacase)and truef∼d,βfalse(xfalse)=const·()1+∥x∥2−β,x∈double-struckRd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula.

Let X 1 , . . . , X N , N > n, be independent random points in R n , distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension n tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

Let X 1 , . . . , X n be independent random points that are distributed according to a probability measure on R d and let P n be the random convex hull generated by X 1 , . . . , X n (n ≥ d + 1). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of P n is strictly monotonically increasing in n.

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