An “Isomorphic” Version of Dvoretzky’s Theorem, II

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 d(Y, `2) ≤ K √ k log(1 + n/k) . In particular , if log n ≤ k ≤ n1−Kε2 , there exists a k-dimensional subspace Y (of an arbitrary n-dimensional normed space X) with d(Y, `2) ≤ K ε √ k log n . 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 [Gl1] and the remark following the proof of Theorem 2 in [Gl2]). In the next lemma we single out what we need from Gluskin’s argument and sketch Gluskin’s proof. Partially supported by BSF and NSF. Part of this research was carried on at MSRI..