Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space

The main result of this article is Theorem 1 which states that a quotient space Y, dim Y = k, of a subspace of any finite dimensional normed space X, dim X — n, may be chosen to be J-isomorphic to a euclidean space even for k = [Xn] for any fixed X < 1 (and d depending on X only). The following theorem is proved. 1. Theorem. For every d > 1 there exists X(d) > 0 such that every n-dimensional normed space X contains a k-dimensional quotient space F of a subspace E C X which satisfies (i) d(F,lj)<d, (ii) dim£= k 3* X(d)n. (Here d(F, lk) denotes a Banach-Mazur distance between two normed spaces; i.e., d(X,Y) = inf{\\T\\-\\T-l\\ over all linear isomorphisms T: X -» Y}.) Moreover, X(d) -» 1 if d -» oo and, for large d, X(d) = 1 — 3-^6 /In In d. Remark 1. It is enough to prove Theorem 1 for large d only, because, as proved in [MJ, any úf-isomorphic copy of l2 contains, for any e > 0, a (1 + e)-isomorphic copy of l2, where k > ic(e)m/d2 and «(e) > 0 depends on e > 0 only. Remark 2. Of course, the theorem states that the dual £ * to £ c X contains a subspace £ * c £ * which satisfies (i) and (ii) of the theorem. Remark 3. In [M2] the theorem was proved with a logarithmic factor, and this theorem was formulated as a problem. We refer the reader to this paper for relevant discussion. 2. Notations. Let Xbe an zi-dimensional normed space, i.e., 7?" with the norm || • ||, and let (x, y) be an inner product on X; consequently, |x| = (x, x)1/2 is a euclidean norm on X. For any x G X let (l/zz)|x| < ||x|| < b\x\ and Mr = jxeS"-'\\x\\ dp(x), where S"_1 = {* € X: |x| = 1} and p(x) = pn_x(x) is the normalized invariant (Haar) measure on S"_1. Let ||*||* = supy^0(\(x, y)\/\\y\\). Then (l/b)\x\ < ||*||* < a\x\, and we define Mr» = js»-i\\x\\*dp(x). Received by the editors July 1, 1984 and, in revised form, August 25, 1984. 1980 Mathematics Subject Classification. Primary 46B20, 46B25, 46C99.