Geometric and probabilistic analysis of convex bodies with unconditional structures, and associated spaces of operators

Geometric and probabilistic analysis of convex bodies with unconditional structures, and... Let $${{\|\cdot\|}}$$ be a norm on $${\mathbb{R}^n}$$ and $${\|.\|_*}$$ be the dual norm. If $${\|\cdot\|}$$ has a normalized 1-symmetric basis $${\{e_i\}_{i=1}^n}$$ then the following inequalities hold: for all $${x,y\in \mathbb{R}^n}$$ , $${\|x\|\cdot\|y\|_*\le \max(\|x\|_1\cdot\|y\|_\infty,\|x\|_\infty\cdot\|y\|_1)}$$ and if the basis is only 1-unconditional and normalized then for all $${x \in \mathbb{R}^n}$$ , $${\|x\|+\|x\|_{*}\leq \|x\|_1+\|x\|_\infty}$$ . We consider other geometric generalizations and apply these results to get, as a special case, estimates on best random embeddings of k-dimensional Hilbert spaces in the spaces of nuclear operators $${{\mathcal N}(K,K)}$$ of dimension n 2, for all k = [λn 2] and 0 < λ < 1. We obtain universal upper bounds independent on the 1-symmetric norm $${\|.\|}$$ for the products of pth moments $$\left( {\mathbb{E}} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|^p\cdot\, \mathbb {E} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|_*^p\right)^{1/p}$$ for independent random variables {f i (ω)}, and 1 ≤ p < ∞. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Geometric and probabilistic analysis of convex bodies with unconditional structures, and associated spaces of operators

, Volume 14 (1) – Mar 17, 2009
22 pages

/lp/springer_journal/geometric-and-probabilistic-analysis-of-convex-bodies-with-N9XwOmLD4i
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0007-6
Publisher site
See Article on Publisher Site

Abstract

Let $${{\|\cdot\|}}$$ be a norm on $${\mathbb{R}^n}$$ and $${\|.\|_*}$$ be the dual norm. If $${\|\cdot\|}$$ has a normalized 1-symmetric basis $${\{e_i\}_{i=1}^n}$$ then the following inequalities hold: for all $${x,y\in \mathbb{R}^n}$$ , $${\|x\|\cdot\|y\|_*\le \max(\|x\|_1\cdot\|y\|_\infty,\|x\|_\infty\cdot\|y\|_1)}$$ and if the basis is only 1-unconditional and normalized then for all $${x \in \mathbb{R}^n}$$ , $${\|x\|+\|x\|_{*}\leq \|x\|_1+\|x\|_\infty}$$ . We consider other geometric generalizations and apply these results to get, as a special case, estimates on best random embeddings of k-dimensional Hilbert spaces in the spaces of nuclear operators $${{\mathcal N}(K,K)}$$ of dimension n 2, for all k = [λn 2] and 0 < λ < 1. We obtain universal upper bounds independent on the 1-symmetric norm $${\|.\|}$$ for the products of pth moments $$\left( {\mathbb{E}} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|^p\cdot\, \mathbb {E} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|_*^p\right)^{1/p}$$ for independent random variables {f i (ω)}, and 1 ≤ p < ∞.

Journal

PositivitySpringer Journals

Published: Mar 17, 2009

References

• The optimal order for the p-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates
Junge, M.

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations