正多胞体的平均宽度与相关高斯变量的期望极大值
概率论
2016-04-07 v3 度量几何
摘要
一个旧猜想陈述：在所有内接于单位球面的单纯形中，正则者具有最大平均宽度。其等价表述为：对任意满足E ξ 1 2 = ⋯ = E ξ n 2 = 1 \mathbb E\xi_1^2= \dots =\mathbb E\xi_n^2=1 E ξ 1 2 = ⋯ = E ξ n 2 = 1 ( ξ 1 , … , ξ n ) (\xi_1,\dots,\xi_n) ( ξ 1 , … , ξ n ) E max { ξ 1 , … , ξ n } ≤ n n − 1 E max { η 1 , … , η n } , \mathbb E\,\max\{\xi_1,\dots,\xi_n\}\leq\sqrt{\frac{n}{n-1}}\, \mathbb E\,\max\{\eta_1,\dots,\eta_n\}, E max { ξ 1 , … , ξ n } ≤ n − 1 n E max { η 1 , … , η n } , η 1 , η 2 , … , \eta_1,\eta_2,\dots, η 1 , η 2 , … , 2 n 2n 2 n 1 ≤ E max { ∣ η 1 ∣ , … , ∣ η n ∣ } E max { η 1 , … , η 2 n } ≤ min { 2 n 2 n − 1 , 1 + C n log n } , 1\leq\frac{\mathbb E\,\max\{|\eta_1|,\dots,|\eta_n|\}}{\mathbb E\,\max\{\eta_1,\dots,\eta_{2n}\}} \leq\min\left\{\sqrt{\frac{2n}{2n-1}}, \, 1+\frac{C}{n\, \log n} \right\}, 1 ≤ E max { η 1 , … , η 2 n } E max { ∣ η 1 ∣ , … , ∣ η n ∣ } ≤ min { 2 n − 1 2 n , 1 + n log n C } , C > 0 C>0 C > 0 W W W ∞ \infty ∞ n n n Q n Q_n Q n W Q n − 2 n π ⟶ n → ∞ d N ( 0 , π − 3 π ) , W_{Q_n} - \sqrt{\frac{2n}{\pi}} \overset{d}{\underset{n\to\infty}\longrightarrow} {\mathcal{N}} \left(0, \frac{\pi-3}{\pi}\right), W Q n − π 2 n n → ∞ ⟶ d N ( 0 , π π − 3 ) ,
引用
@article{arxiv.1511.08479,
title = {Mean width of regular polytopes and expected maxima of correlated Gaussian variables},
author = {Zakhar Kabluchko and Alexander E. Litvak and Dmitry Zaporozhets},
journal= {arXiv preprint arXiv:1511.08479},
year = {2016}
}