ISRAEL JOURNAL OF MATHEMATICS, Vol. 29, Nos. 2-3, 1978 MINKOWSKI SPACES WITH EXTREMAL DISTANCE FROM THE EUCLIDEAN SPACE BY V. D. MILMAN AND H. WOLFSON ABSTRACT It is proved that if the Banach-Mazur distance between an n-dimensional Minkowski space B and l~ satisfies d (Blip) = c',/n (for some constant c > 0 and for big n ) then B contains an A (c)-isomorphic copy of t ~(for k - log log log n). In the special case d(Bfl~)= ",/n, B contains an isometric copy of l~ for k - log n. Introduction We recall that a Minkowski space is a finite dimensional, normed linear space (i.e. a finite dimensional Banach space). The Banach-Mazur distance between two n-dimensional Minkowski spaces B~ and B2 is defined as d = d(Bj, B2)= infr:m~211 Tll II T-' tt, where the infimum is taken over all isomorphisms from B, onto B2. Actually p = log d is a metric on the space of n-dimensional Minkowski spaces but it is more convenient to use d. Clearly d(B~, B2) => 1 and d(B~, B2) = 1 if and only if B~ and B2 are isometric Minkowski spaces. If d(B,, B2) _-< 1 + e we will say that B~ and B2 are e-isometric spaces. F. John [11] proved, that the distance from any n-dimensional Banach space B to l;' is d(B, l~)<= X/n. The maximal distance is attained e.g. for two classical spaces l~, 12: d(lT, l~) = d(12, l;') = ~/n (see [8]). In this paper we will prove two main theorems: THEOREM 1. For every positive integer k there is a positive integer n, such that every n dimensional Banach space B, satisfying d(B,l~) = X/n, contains a k-dimensional subspace Eo C B, which is isometric to lf. Asymptotically (when n "~o~) we have the relation k(n)>= [1/(21n 12)]ln n. Obviously this estimate is exact (up to the coefficient of In n) since 12 contains an I~ with k not greater than log2n. Received April 10, 1977 113