An "Isomorphic" Version of Dvoretzky's Theorem, II

Summary

A different proof is given to the result announced in [MS2]: For each 1 ≤ k < n we give an upper bound on the minimal distance of a k-dimensional subspace of an arbitrary n-dimensional normed space to the Hilbert space of dimension k. The result is best possible up to a multiplicative universal constant.

Our main result is the following extension of Dvoretzky's theorem (from the range 1 < k < c log n to c log n ≤ k < n), first announced in [MS2, Theorem 2]. As is remarked in that paper, except for the absolute constant involved the result is best possible. THEOREM. There exists a K > 0 such that, for every n and every log n : ≤ k < n, any n-dimensional normed space, X, contains a k-dimensional subspace, Y, Satisfying

Jesus Bastero pointed out to us that the proof of the theorem in [MS2] works only in the range k ≤ cn/ log n. Here we give a different proof which corrects this oversight. The main addition is a computation due to E. Gluskin (see the proof of the Theorem in [Gll] and the remark following the proof of Theorem 2 in [G12]). In the next lemma we single out what we need from Gluskin's argument and sketch Gluskin's proof.