The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization
type: Item
instance of: scholarly article
title: The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization
zbMATH Open document ID: 0654.46018
DOI: 10.1007/BF02787120
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(B_ n\) be the set of all n-dimensional Banach spaces and d the Banach-Mazur distance on \(B_ n\). By the result of \textit{F. John} [Courant Anniv. Vol., 187-204 (1948; Zbl 0034.105)], \(d(X,\ell^ n_ 2)\leq n^{1/2}\). The present authors show that \[ \max \{d(X,\ell^ n_{\infty}):X\in B_ n\}=\max \{d(X,\ell^ n_ 1):X\in B_ N\}\leq Cn \exp (-c(\log n)^{1/2}) \] for some (universal) constants \(C,c>0\). The other result (called the proportional Dvoretzky-Rogers factorization theorem) says that for any \(\delta\in (0,1)\) there exists \(A=A(\delta)\) such that, for any finite-dimensional Banach space X, there is an integer \(m>(1-\delta)\dim X\) such that the formal identity \(i_{2,\infty}:\ell^ m_ 2\to \ell^ m_{\infty}\) can be written as \(i_{2,\infty}=\alpha \circ \beta\), \(\beta:\ell^ m_ 2\to X\), \(\alpha:X\to \ell^ m_{\infty}\), with \(\| \alpha \| \cdot \| \beta \| \leq A\). (The same result holds for the formal identity \(i_{1,2}:\ell^ m_ 1\to \ell^ m_ 2.)\)
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A37
zbMATH DE Number: 4067597
zbMATH Keywords: Banach-Mazur distance
zbMATH Keywords: proportional Dvoretzky-Rogers factorization theorem
author: Jean Bourgain
author: Stanislaw J. Szarek
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02787120
OpenAlex ID: W2002139129
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description: scientific article; zbMATH DE number 4067597