transactions of the american mathematical society Volume 294, Number 1, March 1986 ON TYPE OF METRIC SPACES BY J. BOURGAIN, V. MILMAN AND H. WOLFSON1 ABSTRACT. Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed. 1. Introduction. Terminology and notations. 1.1. DEFINITION. Two finite metric spaces (X,p), (Y,d) are called c-isomorphic if there is a one-to-one map ip: X —» Y such that HV'IIlípIIV^Hlíp < c. (We recall that H^IIlip = supx¿y(d(tp(x),ip(y))/p(x,y)). In [G] \\ip\\ ||i/>_1|| is called the distortion of tp. Analogous to the Banach-Mazur distance between normed spaces, we define the Lipschitz distance between finite metric spaces as d(X,Y)= inf I|VHlíp||V'-1||lip, tp : X-+Y where the infimum is taken over all one-to-one and onto maps tp: X —► Y. 1.2. We recall that the Banach-Mazur distance between two n-dimensional normed spaces X,Y is defined as d(X,Y) = inf{||T|| WT'1]]: T: X -> Y is an isomorphism}. It is known that for the finite-dimensional real Banach spaces the Lipschitz distance coincides with the Banach-Mazur distance. (In the nontrivial direction use the fact that for any Lipschitz map i¡> : X —» Y (dim X = dim Y < oo) there is a point x G X such that rp and V-1 have derivatives at x and y = ip(x) respectively. Hence, there is a linear map T: X —► Y satisfying ||T|| ||T||_1 < MlípII^IIlíp.) 1.3. Let Cn = {(£i,..., en)|e¿ G {0,1}} = {0,1}". (Sometimes it is more conve- nient to use C2 = {-1,1}™. It will be clear from the context what representation is used.) For every pair e = (£y)"=1, e' = (£y)™=1 in C2 the Hamming metric is defined as h(e,e') = #{¿|e¿ ^ e[}. In the case of C2 = {0,1}" this metric coincides with the standard If metric Q3"=1 |e¿ — e¿|). DEFINITION. Let 1 < p < oo. The metric space (C%,pp), where C% = {0,1}™ and Pp(e,e') = [£JL, \e>- ejlf/p = [^=1 \£i - e'^ for any pair e,e' G C? is called the lp n-cube (or lp-cube). Received by the editors May 25, 1984. 1980 Mathematics Subject Classification. Primary 46B20, 47H99, 51E99, 54E40. xThis paper includes part of the Ph.D. thesis of the third author. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 295 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use