The random version of Dvoretzky s theorem in l n

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1 The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the Euclidea ball i this sectio. We also show that, up to a absolute costat the estimate o k caot be improved. Itroductio ilma s versio of Dvorezky s theorem states that: There is a fuctio cε) > 0 such that for all k cε) log, l k + ε)-embeds ito ay ormed space of dimesio. See [Dv] for the origial theorem of Dvoretzky i which the depedece of k o is weaker), [i] for ilma s origial work, ad [S] ad [Pi] for expository outlets of the subject there are may others). It would be importat for us to otice that the proofs) of the theorem above actually give more: The vast majority of subspaces of the stated dimesio are + ε)-isomorphic to l k. The depedece of k o i the theorem above is kow to be best possible for l ) but the depedece o ε is far from beig uderstood. The best kow estimate is cε) cε log ε ) give i [Sc] here ad elsewhere i this paper c ad C deote positive uiversal costats). However, the proof i [Sc] does ot give the additioal iformatio that most subspaces are + ε)-isomorphic to l k. If oe also wat this requiremet the the best estimate for cε) that was kow was cε) cε [Go]). As a upper boud for cε) oe gets C/ log ε for some uiversal C. Ideed, if lk + ε) embed ito l the k C log / log ε. This is also the right order of k i the l case: If k c log / log ε the lk + ε) embed ito l. We show here that, i the l case, if oe is iterested i the probabilistic statemet of Dvoretzky theorem i.e, that the vast majority of subspaces of l of a certai dimesio are + ε)-isomorphic to Euclidea spaces) the the right estimate for cε) is cε. Supported by the Israel Sciece Foudatio

2 Theorem For k < cε log, with probability > e ck, the l orm ad a multiple of the l orm are + ε equivalet o a k dimesioal subspace. oreover, this does t hold aymore for k of higher order. i.e., For every a there is a A such that if, with probability larger tha e ak, a k dimesioal subspace satisfies that the ratio betwee the l orm ad a multiple of the l orm are + ε equivalet for all vectors i the subspace, the k Aε log. Computatio of the cocetratio of the max orm Let g, g,... be a sequece of stadard idepedet Gaussai variables. fix ad let be the media of g, g,..., g ). I this sectio we compute some fie estimates o the probability of deviatio of g, g,..., g ) from. Claim Proof: Cosequetly, or / ) + ) e / / ) e e ). ) P max g i < ) i / e s / ds + + e / e e ), e s / ds). + se s / ds + ) e / / ) e e ). 3) ) Similarly, or / e s / ds se s / ds e /, e / / ). 4) Claim log 4 + log e 3ε / P max i g i > + ε)) log + o))e ε 5) where o) meas a) with a) 0 as idepedetly of ε.

3 Proof: 3) implies P max i g i > + ε)) / +ε) e +ε) +ε) / + ) e e e ε e ε / 6) ad, sice is of order log, we get from this that P max i g i > + ε)) log + o))e ε 7) For a fixed ε oe ca replace log + o)) with a quatity tedig to 0 with.) We ow look for a lower boud o P max i g i > + ε)). Sice for iid X i -s, P max i X i > t) P X > t)) e P X >t) P X >t), 8) +P X >t) +P X >t) P max i g i > + ε)) P g > + ε)) + P g > + ε)). 9) The right had side is a icreasig fuctio of P g > + ε)) ad, by 4), P g > + ε)) / +ε)+ e +ε) e e +ε) ) +ε)+ e / e ε ε / e e +ε) ) / ) +ε)+ e ε ε / e e +ε) ) log 4 e ε ε / log 4 e 3ε /, for ε ad large eough idepedetly of ε). Usig 9), we get P max i g i > + ε)) 0) log 4 e 3ε / + log log 4 e 3ε / 4 + log e 3ε /. ) Claim 3 For some absolute positive costats c, C ad for all 0 < ε < /, exp Ce ε ) P max i g i < ε)) C exp ce 3ε /4 ) ) Proof: P max i g i < ε)) ε) ε) ) e s / ) e s / 3

4 ) se s / ε) e ε) / e / ) ) ) e / e ε ε / ) / )e ε ε / )) by4) exp / )e ε ε / )) + o)) exp log + o))e 3ε /4 ) Which proves the right had side iequality i ). As for the left had side, ) P max g i < ε)) e s / i ε) ε) ε) ε) e ε) / e / ε) eε ε / ) se s / exp / ) + e eε ε / ) by3) exp log + o))e ε ). ) ) We summarize Claims ad 3 i a form that will be useful for us later i the followig Propositio. Propositio For some positive absolute costats c, C ad for all 0 < ε < ad N, deotig g g, g,..., g ), ce Cε log P g < ε) g or g > + ε) g ) Ce cε log. Proof: This follows easily from Claims ad 3 ad the facts that e x > x for all x, is of order log ad P g < ε) or g > + ε) ) < Ce ε. 4

5 3 Proof of the theorem The first part of the Theorem follows easily from the, by ow well exposed, proof of ilma s versio of Dvorezky s theorem see e.g, [S] or [Pi]) with the improved cocetratio estimate i the right had side of the iequality i) Propositio replacig the classical estimates. For the proof of the secod part we eed: Lemma Let A be a subset of G,k of µ,k measure a. Put U A E A E, the P g, g,..., g ) U A ) a /k. Proof: Let X, X,..., X k be k idepedet radom vectors distributed accordig to P, the caoical Gaussia measure o R. Note that, sice µ,k is the uique rotatioal ivariat probability measure o G,k, the distributio of spa{x,..., X k } is µ,k. Accordigly, P U A ) k P X, X,..., X k U A ) P spa{x, X,..., X k } A) µ,k A). Remark As we ll see below we use oly a weak form of Lemma. We actually believe there is a much stroger form of it. Proof of the moreover part i Theorem : Let A G,k be such that every E A there is a E such that E x x + ε) E x for all x E. Let B be the subset of A of all E for which 3ε) let C A \ B. By Lemma, µ,k C) /k P {x; x < + ε) 3ε) x or x > E +ε), ad + ε) x }) ad, by Propositio, this last quatity is smaller tha Ce cε log. It follows that µ,k B) > e ak Ce cεk log. We may assume that ε log is much larger tha a so that the last term above is domiated by e ak. Applyig Lemma oce more we get P {x; 3ε) x x + ε) x }) µ,k B) > e ak. Usig ow the other part of Propositio we get that Cε log > ak. 5

6 Refereces [Dv] Dvoretzky, A., Some results o covex bodies ad Baach spaces. 96 Proc. Iterat. Sympos. Liear Spaces Jerusalem, 960) pp Jerusalem Academic Press, Jerusalem; Pergamo, Oxford. [Go] Gordo, Y., Some iequalities for Gaussia processes ad applicatios. Israel J. ath ), o. 4, [i] ilma, V. D., A ew proof of A. Dvoretzky s theorem o cross-sectios of covex bodies. Russia) Fukcioal. Aal. i Prilože. 5 97), o. 4, [S] ilma, V. D. ad Schechtma, G., Asymptotic theory of fiite-dimesioal ormed spaces, Lecture Notes i athematics, 00, Spriger-Verlag, Berli, 986. [Pi] Pisier, G., The volume of covex bodies ad Baach space geometry. Cambridge Tracts i athematics, 94. Cambridge Uiversity Press, Cambridge, 989. [Sc] Schechtma, G., Two observatios regardig embeddig subsets of Euclidea spaces i ormed spaces, Advaces i ath., to appear. Gideo Schechtma Departmet of athematics Weizma Istitute of Sciece Rehovot, Israel gideo.schechtma@weizma.ac.il 6