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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.