Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov. "Volume thresholds for Gaussian and spherical random polytopes and their duals." Studia Mathematica 183.1 (2007): 15-34. <http://eudml.org/doc/284425>.

@article{PeterPivovarov2007,
abstract = {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 $V_\{N\}: = vol(K_\{N\} ∩ RB₂ⁿ)/vol(RB₂ⁿ)$. 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.},
author = {Peter Pivovarov},
journal = {Studia Mathematica},
keywords = {high-dimensional convex bodies; random polytopes},
language = {eng},
number = {1},
pages = {15-34},
title = {Volume thresholds for Gaussian and spherical random polytopes and their duals},
url = {http://eudml.org/doc/284425},
volume = {183},
year = {2007},
}

TY - JOUR
AU - Peter Pivovarov
TI - Volume thresholds for Gaussian and spherical random polytopes and their duals
JO - Studia Mathematica
PY - 2007
VL - 183
IS - 1
SP - 15
EP - 34
AB - 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 $V_{N}: = vol(K_{N} ∩ RB₂ⁿ)/vol(RB₂ⁿ)$. 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.
LA - eng
KW - high-dimensional convex bodies; random polytopes
UR - http://eudml.org/doc/284425
ER -