Random Euclidean sections of some classical Banach spaces

Abstract

Using probabilistic arguments, we give precise estimates of the Banach-Mazur distance of subspaces of the classical $\ell_q^n$ spaces and of Schatten classes of operators $S_q^n$ for $q \ge 2$ to the Euclidean space. We also estimate volume ratios of random subspaces of a normed space with respect to subspaces of quotients of $\ell_q$. Finally, the preceeding methods are applied to give estimates of Gelfand numbers of some linear operators.