RANDOM EUCLIDEAN SECTIONS OF SOME CLASSICAL BANACH SPACES

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