Expected intrinsic volumes and facet numbers of random beta‐polytopes

By Zakhar Kabluchko, Daniel Temesvari, C. Thaele. Mathematische Nachrichten, 2017

52 citations recorded by the source corpus.

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,β(x)=const·1−∥x∥2β,∥x∥<1,(thebetacase)and f∼d,β(x)=const·1+∥x∥2−β,x∈Rd,(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.

Subfields: Applied Mathematics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/a514f9e22ab0d8a8c2ddf8cc70de7b213e92ec91

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In Applied Mathematics

1. Zakhar Kabluchko
2. Dmitry Zaporozhets
3. C. Thaele
4. Matthias Schulte
5. Günter Last
6. Giovanni Peccati

In Molecular Biology

1. Dmitry Zaporozhets
2. C. Thaele
3. Zakhar Kabluchko

Citations in the corpus

• Beta polytopes and Poisson polyhedra: f-vectors and angles (2018, methodology): “For faces of maximal dimension it was established in the work of Beermann and Reitzner [7, 8] for Gaussian polytopes and in [9] by Bonnet, Grote, Temesvari, Thäle, Turchi and Wespi for beta and beta’ polytopes.”