Subspaces of \(\ell^ N_ p\) of small codimension (Q1802759)

Let \(1\leq p\leq\infty\) and \(m\) and \(N\) integers. For what numbers \(k\) do there exist good copies of \(\ell^ k_ p\) in \(X\) for arbitrary \(m\)- dimensional subspaces \(X\) of \(\ell^ k_ p\)? In the introduction one finds a thorough summary of results concerning this question. Whereas previous work mainly has concentrated on the case of small \(m\) here the case \(m= N- n\) with small \(n\) is of interest. A typical result reads as follows: Let \(E\) be a subspace of \(\ell^ N_ 1\) of dimension \(N- n\). Then \(E\) contains a \(k\)-dimensional subspace \(F\) such that \(d(F,\ell^ k_ 1)\leq 6\), where \[ k\geq (1/24)^ 2\min\bigl\{(N/4n)\log(N/2N),N\bigr\}. \] The estimates in the case \(p=\infty\) are in terms of the Gelfand numbers of the formal identity between \(\ell^ N_ 1\) and \(\ell^ N_ \infty\).