, Volume 14, Issue 1, pp 83–104 | Cite as

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



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 < ∞.


Local theory of normed spaces Symmetric and unconditional bases Convex bodies Nuclear spaces Random Gaussian operators 

Mathematics Subject Classification (2000)

46B03 46B07 46B09 46B20 46B28 


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© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsTechnionHaifaIsrael
  2. 2.Equipe d’Analyse et de Mathématiques AppliquéesUniversité de Paris-Est-Marne-la-ValléeMarne-la-Vallée Cedex 2France

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