Optimal subspaces for n-widths of p-ellipsoids (Q801494)

Let \(E=\{x\in \ell^ N_ p;\sum^{N}_{i=0}| x_ i/a_ i|^ p\leq 1\}\) where \(1\leq p\leq \infty\); \(\{a_ i\}^ N_{i=0}\) (N\(\leq \infty)\) is a strictly decreasing, sequence of positive scalars. It is known that for the n-width of E (as defined by Kolmogorov) we have \(d_ n(E,\ell_ p)=a_ n\) (n\(\geq 0)\). The set of subspaces \(L_ n\) of those n-dimensional subspaces in \(\ell_ p\) for which such value is attained, is studied here. More precisely, it is shown what perturbations of the subspace \(sp\{e_ i\}^{n-1}_{i=0}\) still give ''optimal'' subspaces. These results generalize what for ellipsoids in Hilbert spaces had been done by \textit{R. S. Ismagilov} [Geometry of linear spaces and operator theory, interoff. thematic Work Collect., Yaroslav, 1977, 75-113 (1977; Zbl 0418.41017)].