Expected intrinsic volumes and facet numbers of random beta-polytopes

Abstract

Let $X_1,\dots ,X_n$ be i.i.d.\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$f_{d,\beta }(x)=\text{const}\cdot (1-\Vert x{\Vert }^2{)}^{\beta },\Vert x\Vert \le 1,\text{(the beta case)}$$ and $${\tilde{f}}_{d,\beta }(x)=\text{const}\cdot (1+\Vert x{\Vert }^2{)}^{-\beta },x\in {\mathbb{R}}^d,\text{(the beta’ case).}$$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\dots ,X_n$. Asymptotic formulae where obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when $\beta \downarrow -1$, respectively $\beta \uparrow +\infty$, we can also cover the models in which $X_1,\dots ,X_n$ 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 $\pm X_1,\dots ,\pm X_n$ and $0,X_1,\dots ,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.