Expected intrinsic volumes and facet numbers of random beta-polytopes

Abstract

Let $X_1,\ldots,X_n$ be i.i.d.\ random points in the $d$-dimensional Euclidean space sampled according to one of the following probability densities: $$ f_{d,\beta} (x) = \text{const} \cdot (1-\|x\|^2)^{\beta}, \quad \|x\|\leq 1, \quad \text{(the beta case)} $$ and $$ \tilde f_{d,\beta} (x) = \text{const} \cdot (1+\|x\|^2)^{-\beta}, \quad x\in\mathbb{R}^d, \quad \text{(the beta' case).} $$ We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of $X_1,\ldots,X_n$. 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 $X_1,\ldots,X_n$ 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 $\pm X_1,\ldots,\pm X_n$ and $0,X_1,\ldots,X_n$. One of the main tools used in the proofs is the Blaschke-Petkantschin formula.