The dimension of almost spherical sections of convex bodies

The well-known theorem of Dvoretzky 1 states that convex bodies of high dimension have low dimensional sections which are almost spherical. More precisely, the theorem states that for every integer k and every e 0 there is an integer n (k, e) such that any Banach space X with dimension n (k, e) has a subspace Y of dimension k with d{Y, l\\) 1 + e. Here d (Y, l£) denotes the Banach Mazur distance coefficient between Y and the k dimensional Hubert space l\\ i. e. inflim \\\ ~ X \\ \\ taken over all operators T from Y onto l\\. The estimate for n (k, e) given in 1 was improved in 5 to n (k, e) = ec (e) k In other words (considering the dependence of n (k, e) on k for fixed e) the dimension of the almost spherical section (of the unit ball) given by Dvoretzkys theorem is about the log of the dimension of the space. This estimate is in general the best possible, since as observed in 10 it is easy to verify that if X = Z ~ any subspace Y of X whose Banach Mazur distance from a Hubert space is 2 say, must be of dimension at most C log n. It turns out however that if