Combinatorics

Expected value

Euclidean space

Convex hull

Cover (algebra)

Mathematics

Hull

Statistics

Mathematical proof

Regular polygon

Engineering

Convex body

Polytope

Marine engineering

Unit sphere

Geometry

Mechanical engineering

Mathematische Nachrichten

Let be i.i.d. random points in the d ‐dimensional Euclidean space sampled according to one of the following probability densities: urn:x-wiley:0025584X:media:mana201700255:mana201700255-math-0002 and urn:x-wiley:0025584X:media:mana201700255:mana201700255-math-0003 We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of . 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 , respectively , we can also cover the models in which 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 and . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.

Expected intrinsic volumes and facet numbers of random beta‐polytopes