The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization
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 \(i_{1,2}:\ell^ m_ 1\to \ell^ m_ 2.)\)
proportional Dvoretzky-Rogers factorization theorem
scientific article; zbMATH DE number 4067597
scientific article