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• # Artykuł - szczegóły

## Studia Mathematica

2007 | 183 | 1 | 15-34

## Volume thresholds for Gaussian and spherical random polytopes and their duals

EN

### Abstrakty

EN
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.

15-34

wydano
2007

### Twórcy

autor
• Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1