Expected intrinsic volumes and facet numbers of random beta‐polytopes

LetX 1 , ⋯ , X nbe i.i.d. random points in the d ‐dimensional Euclidean space sampled according to one of the following probability densities:f d , β( x ) = const ·1 − ∥ x ∥ 2 β ,∥ x ∥ < 1 , (the beta case) andf ∼ d , β( x ) = const ·1 + ∥ x ∥ 2− β , x ∈ R d , (the beta" case). We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull ofX 1 , ⋯ , X n . 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 whichX 1 , ⋯ , X nare uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ± X 1 , ⋯ , ± X nand 0 , X 1 , ⋯ , X n . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.