Exact evaluation of Kolmogorov diameters of finite-dimensional octahedra (Q909195)

Let X be a Banach space with unit sphere B. Let m be a positive integer and \(S\subset X\) a set symmetric by the origin. The Kolmogorov diameter \(d_ m(S,X)\) is defined as the infimum of all \(\epsilon >0\) such that \(S\subset L+\epsilon B\) for some m-dimensional subspace L of X. Let \(\ell^ n_ p\) mean the n-dimensional \(\ell_ p\) space, \(1\leq p\leq +\infty\), and denote by \(B^ n_ p\) its unit ball. A computation, or at least an estimate of the numbers \(d_ m(B^ n_ p,\ell^ n_ q)\) represents an interesting geometrical question. There exist in the literature several results in this direction. In this paper under review two new facts are proved: \(d_{3k-2}(B_ 1^{3k},\ell_{\infty}^{3k})=1/2k\) and \(d_{n-k}(B^ n_ 1,\ell^ n_{\infty})=3/2n+o(n^{-1}),\) where \(o(n^{-1})\geq 0\), for every positive integer k.