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Expected intrinsic volumes and facet numbers of random beta‐polytopes
Author(s) -
Kabluchko Zakhar,
Temesvari Daniel,
Thäle Christoph
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700255
Subject(s) - mathematics , polytope , convex hull , combinatorics , regular polygon , cover (algebra) , mathematical proof , hull , euclidean space , facet (psychology) , euclidean geometry , convex body , geometry , mechanical engineering , marine engineering , engineering , psychology , social psychology , personality , big five personality traits
LetX 1 , ⋯ , X nbe i.i.d. random points in the d ‐dimensional Euclidean space sampled according to one of the following probability densities:f d , β( x ) = const ·1 − ∥ x ∥ 2 β ,∥ x ∥ < 1 , (the beta case) andf ∼ d , β( x ) = const ·1 + ∥ x ∥ 2− β , x ∈ R d , (the beta" case). We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull ofX 1 , ⋯ , X n . 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 whichX 1 , ⋯ , X nare uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ± X 1 , ⋯ , ± X nand 0 , X 1 , ⋯ , X n . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.