AN \ ISOMORPHIC " VERSION OF DVORETZKY ' S THEOREM , IIby

A diierent 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) rst announced in MS2, Theorem2]. As is remarked in MS2], except for the absolute constant involved the result is best possible. K q k log(1+n=k) : In particular, if log n k n 1?K" 2 , there exists a k-dimensional subspace Y (of an arbitrary n-dimensional normed space X) with d(Y; ` k 2) K " q 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 diierent 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.