Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov

Peter Pivovarov

Studia Mathematica, Tome 178 (2007), p. 15-34 / Harvested from The Polish Digital Mathematics Library

Résumé

Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_{N}$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_{N} : = v o l (K_{N} \cap R B ₂ ⁿ) / v o l (R B ₂ ⁿ)$ . For a large range of R = R(n), we establish a sharp threshold for N, above which $V_{N} \to 1$ as n → ∞, and below which $V_{N} \to 0$ as n → ∞. We also consider the case when $K_{N}$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both R ∈ (0,1) and R = 1. Lastly, we prove complementary results for polytopes generated by random facets.

Publié le : 2007-01-01

Zbl 1134.52005

EUDML-ID : urn:eudml:doc:284425

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     author = {Peter Pivovarov},
     title = {Volume thresholds for Gaussian and spherical random polytopes and their duals},
     journal = {Studia Mathematica},
     volume = {178},
     year = {2007},
     pages = {15-34},
     zbl = {1134.52005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-2}
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Peter Pivovarov. Volume thresholds for Gaussian and spherical random polytopes and their duals. Studia Mathematica, Tome 178 (2007) pp. 15-34. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-2/