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A remark concerning the dependence on \(\epsilon\) in Dvoretzky’s theorem. (English) Zbl 0679.46011
Geometric aspects of functional analysis, Isr. Semin., GAFA, Isr. 1987-88, Lect. Notes Math. 1376, 274-277 (1989).
[For the entire collection see Zbl 0668.00010.]
The main result of this paper is the following Theorem 2.
Let \(\| \cdot \|\) be a norm on \({\mathbb{R}}^ N\). Let \(\sigma\) be the norm of the identity map from \(\ell^ N_ 2\) to \(({\mathbb{R}}^ N,\| \cdot \|)\). Then for every \(\epsilon >0\) there exists \(n\in {\mathbb{N}}\) with \[ n\geq c\cdot \epsilon^ 2\cdot ({\mathbb{E}}\| \sum^{N}_{i=1}g_ ie_ i\| /\sigma)^ 2 \] and an operator T: \(\ell^ n_ 2\to^{onto}Y\subseteq ({\mathbb{R}}^ N,\| \cdot \|)\) for which \(\| T\| \cdot \| T^{-1}\| \leq 1+\epsilon.\)
Here \(\{g_ i\}^ N_ 1\) is a sequence of independent standard Gaussian variables, \(c>0\) is an absolute constant.
Reviewer: M.I.Kadets

46B20 Geometry and structure of normed linear spaces