Expected intrinsic volumes and facet numbers of random beta‐polytopes

Abstract

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.

Threshold phenomena for high-dimensional random polytopes

“…As pointed out in [11], the expected k-th intrinsic volume of P β N,n is directly connected to the expected k-dimensional volume of P α N,k for some different parameter α depending on β, k and n. Because of this, Theorem 1.1 can be applied to establish threshold results for the intrinsic volumes V k (P β N,n ), k ∈ {1, . .…”

“…Lately, the high-dimensional geometry of sets arising from these models of randomness have been studied extensively; for instance, in terms of properties of their volume [10], facet numbers [5] or intrinsic volumes [11]. Asymptotic estimates on the expected volume of the polytope P β N,n , as N → ∞, were derived by Affentranger [1] for any fixed dimension n and parameter β.…”

“…For the beta distribution, there is a known formula that relates the expected k-th intrinsic volume of P β N,n to the expected volume of the respective k-dimensional polytope up to a different parameter β ′ . In particular, Proposition 2.3 in [11] states that…”

“…Similarly as before, we use the upper bounds in (13), Lemma 4.4 (a) and (11) to see that E V n (H β N,n ∩ sB n 2 ) V n (sB n 2 ) t s n + exp(−N F (t −1 )).…”

Lyapunov Exponents for Some Isotropic Random Matrix Ensembles

“…, N) of (2.5) reclaims (1.11). Also, we remark that random polytopes formed by the convex hull of the columns of X T , in the case that X is of size n × N (n > N) with rows chosen according to one of the distributions (2.1), have been the subject of the recent studies [18,19].…”