ESTIMATES OF THE WEAK DISTANCE BETWEEN FINITE-DIMENSIONAL BANACH-SPACES

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for every n-dimensional Banach spaces X and Y a measure s pace Omega and two operator-valued functions T: Omega --> L(X,Y), S: O mega --> L(Y, X) so that integral(Omega)S(omega)oT(omega) d omega is t he identity operator in X and integral(Omega)parallel to S(omega)paral lel to .parallel to T(omega)parallel to d omega = O(n(alpha)) for some absolute constant alpha < 1. We prove also that any subset of the uni t n-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isom orphic to a k-cube.