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

Loading next page...
 
/lp/springer_journal/geometric-and-probabilistic-analysis-of-convex-bodies-with-N9XwOmLD4i
Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
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.

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

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.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off