Volume thresholds for Gaussian and spherical random polytopes and their duals

• Autorzy: Peter Pivovarov
• Języki publikacji: 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.

Kategorie tematyczne

• 52A22: Random convex sets and integral geometry
• 52A38: Length, area, volume
• 52B11: n -dimensional polytopes

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 183
• Numer: 1
• Strony: 15-34

Daty

wydano

2007

Twórcy

autor

Peter Pivovarov

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

Identyfikatory

DOI

10.4064/sm183-1-2