# 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
Springer Journals
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.

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