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Volume thresholds for Gaussian and spherical random polytopes and their duals

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Pivovarov P.

Studia Mathematica

2007

tom 183

nr 1

15-34

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 $V_{N}: = 𝔼 vol(K_{N} ∩ RB₂ⁿ)/vol(RB₂ⁿ)$. For a large range of R = R(n), we establish a sharp threshold for N, above which $V_{N} → 1$ as n → ∞, and below which $V_{N} → 0$ 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.

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1 Studia Mathematica

1 Pivovarov P.

1 2007

Wyniki wyszukiwania

Volume thresholds for Gaussian and spherical random polytopes and their duals