Dedicated to the Memory of Jose Luis Rubio De Francia

In this paperwe prove that the /~.,-cube can be (1 + s)-embedded into any 1 -subsyntmetrie C(s>n.dimensional normed space. Marcus and Pisier in [5]iniciated tite study of tite geometry ob finite metric spaces. Bourgain, Milman and Wolbson introduced a new notion of metnc type and developed tite non-linear titeory of Banacit spaces (see [2]and [7]). AII titese themes have been studied more intensively over tite last years. Soitnson and Lindenstrauss proved that, given N points in tite Euclidean space, titey can be (1 + s)-embedded into a subspace of dimension Kfr) /og N (see lemma 1 in [3]). Tite metitod titey use is based in tite isoperimetrie inequality ob P. Levy. Anotiter proof of tite same fact was given by Pisier, using Gaussian processes ([8]). Bourgain, Milman and Wolfson, in tite paper before mentioned, studied tite /;-cubes and titeir (1 + s)-embeddings in finite metric spaces. More recently, Scitecittman obtained estimates bor (1 + s)-embeddings of finite subsets of Lr into /;-spaces (see [9]). In titis paper we will study (1 + c)-embeddings of tite ft-cube in finite-dimensional subsymmetric spaces. The result we prove for tite 4-case 1 =p< 2, can be deducedbrom Joitnson and Lindenstrauss’s lemma plus arefinement of Dvoretzky’s titeorem (see for instance [7],Theorem 3.9), but, as far as we know, it is new in otiter cases. The metitod we use is in essence of probabilistic nature and tite main tool is a well known deviation inequality. * Supported by CAICYT 0804-84 ** Supported by DGA (Spain> Supported in pan by NSF 1980 Mathematics Subject Classification (1985 revision): 46H20 Editorial de la Universidad Complutense. Madrid, 1989. http://dx.doi.org/10.5209/rev_REMA.1989.v2.18062 48 J. Bastero-]. Bernues-N. Kalton We begin by recalling some definitions. Given twometile spaces (M,d) and (Md’), we say titat (M,d) (1 ±s)-embedsinto (M’,d’) iftitere isa one-to-one mapffrom Minto M’ such that ¡[f¡L~¡Lf”’¡I~, =1±8, witere ¡¡fIL. = sup~,~ d(xy) Tite !L-cube is the metric space (Q, p,.,) witere C~ = (— 1, + 1>’ and pjs,s’)= máx s, — s’I, for any par of elements s,g’ belonging to C;. Since p.,(c,e’) = 2, whenever s # 8’, the problem we are consideiing may be related witit the sphere-packing problem, i.e., itow many balís, witit radius l—s 2 can be packed into tite unit balí of a finite dimensional Banacit space, in an asymptotic way? (See the paper by BalI [1] for inbtnite dimensional sphere-packing problem) In tite sequel E., wilI denote a finite-dimensional Banacit space witit a l-subsymmetric normalized basis (e,...,e.,}. We use standard Banacit space theory notation as may be bound in [4]. Tite theorem we will prove itere is tite following Theorem.—Tirere exists a nunierical constaní C> O sucir ihat, for any s> O we can jind a subsel ofNpoints {Xn~..,irN> in E,, verzfr¡ng I—s<¡Ix—x,I¡=l+s, ¡ ti provided that