Symmetric block bases in finite-dimensional normed spaces
It is shown that for \(1\leq p<\infty\) and \(\epsilon\in]0,[\) any basis of \(\ell^ n_ p\), which is C-equivalent to the unit vector basis of \(\ell^ n_ p\) has a \((1+\epsilon)\)-symmetric block basis of cardinality proportional to n/log n, where the proportionality factor only depends on C, \(\epsilon\) and p. When \(1<p<\infty\), an upper bound proportional to n log log n/log n of such \((1+\epsilon)\)-symmetric block bases is also obtained. The proofs are based on measure theoretic methods and the results generalize results of \textit{D. Amir} and \textit{V. D. Milman} [Isr. J. Math. 50, 1-12 (1985; Zbl 0612.46021)].
scientific article; zbMATH DE number 4180087