Estimates of the weak distance between finite-dimensional Banach spaces

Abstract

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫Ω S(ω)oT(ω)dω is the identity operator inX and ∫Ω||S(ω)||·||T(ω)||dω=O(n α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube.