The distance of a subspace of \(R^ m\) from its axes and n-widths of octahedra (Q760866)

Necessary and sufficient conditions for the existence of an n-dimensional subspace of \(R^ m\) with given \(\ell^ m_ p\) distances from the principal axes are established, and used to derive exact results and estimates for the n-widths of diagonal operators from \(\ell^ m_ 1\) to \(\ell^ m_ p\). In particular, evidence is presented to support the conjecture that the asymptotic value of \(d_ n(I;\ell^ m_ 1,\ell^ m_ p)\) is \(m^{1/p}\sqrt{1-\alpha}/\sqrt{n}\) where \(\alpha =\lim n/m\).