AN \ISOMORPHIC" VERSION OF DVORETZKY'S THEOREM, II by Vitali D. Milma ad Gideo Schechtma Abstact - A dieet poof is give to the esult aouced i [MS2]: Fo each <we give a uppe boud o the miimal distace of a -dimesioal subspace of a abitay -dimesioal omed space to the Hilbet space of dimesio. The esult is best possible up to a multiplicative uivesal costat. Ou mai esult is the followig extesio of Dvoetzy's theoem (fom the age <<clog to c log <) st aouced i [MS2, Theoem2]. As is emaed i [MS2], except fo the absolute costat ivolved the esult is best possible. Theoem. Thee exists a K> such that, fo evey ad evey log <,ay - dimesioal omed space, X, cotais a -dimesioal subspace, Y, satisfyig d(y `2) K log(+=) : I paticula, if log ;K"2, thee exists a -dimesioal subspace Y (of a abitay -dimesioal omed space X) withd(y `2) K " log : Jesus Basteo poited out to us that the poof of the theoem i [MS2] wos oly i the age c= log. Hee we give a dieet poof which coects this ovesight. The mai additio is a computatio due to E. Glusi (see the poof of the Theoem i [Gl] ad the ema followig the poof of Theoem 2 i [Gl2]). I the ext lemma we sigle out what we eed fom Glusi's agumet ad setch Glusi's poof. Glusi's Lemma. Let =2 ad let deote the omalized Haa measue o the Gasmaia of -dimesioal subspaces of IR. The, fo some absolute positive costat c, (E 9x 2 E with x <c log( + =) x 2 < =2: Poof. Let g i j, i = ::: j = :::, be idepedet stadad Gaussia vaiables. The ivaiace of the Gaussia measue ude the othogoal goup implies that the coclusio of the lemma is euivalet to Pob if x2s ; max j j P x ig i j j P log( + =) ( (P x l= ig i l ) 2 ) <c =2! < =2: Patially suppoted by BSF ad NSF. Pat of this eseach was caied o at MSRI.
As is well ow the vaiable ( P j= (P x ig i j ) 2 ) =2 is well cocetated ea the costat x 2 =( P Pob x2 i )=2p. I paticula, @ 9x 2 S ; with ( X j= ( X if is lage eough. It is thus eough to pove that Pob if x2s ; X max j j The left had side hee is clealy domiated by Pob X 2 @ if x2s ;( j= j X x i g i j ) 2 ) =2 > 2 p A < =4 x i g i j j < 2c p log( + =)! < =4: x i g i j j 2 ) =2 < 2c p 2 log( + =) whee fa i g deotes the deceasig eaagemet of the seuece fja i jg. To estimate the last pobability we use the usual deviatio ieualities fo Lipschitz fuctios of Gaussia vectos (see e.g. [MS] o [Pi]). The oly two facts oe should otice ae that the om )=2 o IR is domiated by the Euclidea om, i.e. that the Lipschitz a =( P 2 j= a2 j costat of the fuctio a =( P 2 j= a2 j A )=2 o IR is at most oe ad that the expectatio of (g ::: g ) is lage tha c (2) =2p log( + =) fo some absolute costat c. The, fo a appopiate c, Pob X 2 @ if x2s ;( j= if is lage eough. j X x i g i j j 2 ) =2 < 2c p 2 log( + =) A < exp(; log(+=)) < =4 Poof of the theoem. The poof follows i pats the poof i [MS2]. Fo completeess we shall epeat these pats. I what follows < K<deote absolute costats, ot ecessaily the same i each istace. By a esult of Bougai ad Szae (Theoem 2 of [BS] but efe to ema 4 i [MS2] fo a explaatio why we eed oly a much simple fom of thei esult), we may assume without loss of geeality that thee exists a fo all x 2 Z fo subspace, Z X, withm = dimz > =2 ad x`m x Z x`m 2 some absolute costat >. Let M deote the media of x Z ove S m; = S ; \ Z. Fix as i the statemet of the theoem. If M>K the, by [Mi] (see also [FLM] o [MS], Theoem 4.2), Z ad thus X cotais a -dimesioal subspace, Y, satisfyig d(y `2) 2. Also, fo each, with pobability > ; e ; () x2 M + K 2! x 2 :
This agai follows by the usual deviatio ieualities. Let us efesh the eade's memoy: Let M be a 2 -et i the sphee of a xed -dimesioal subspace, Y,ofZ with jmj 6 (see [MS], Lemma 2.6). Deotig by the Haa measue o the othogoal goup O(m) we get, ( U Ux Z M + K fo some x 2M exp( log 6 ; K 2 ): Thus, a successiveappoximatio agumetgives that, with pobability lage tha ; e ;, a -dimesioal subspace E of Z satises! x2 M + K x 2 fo all x 2 E: By Glusi's Lemma, fo <=4, m (E 9x 2 E with x <c m (E 9x 2 E with x <c log( + =) log( + m=) m x 2 x 2 < =2: That is, with pobability lage tha =2 a-dimesioal subspace E satises log( + =) xc x 2 fo all x 2 E: Combiig this with (), we get that, if M K whose distace to Euclidea space is smalle tha 4K log( + =) c, thee exists a -dimesioal subspace = K p =log( + =): Rema. Ca oe show that the coclusio of the Theoem holds fo a adom subspace Y? The oly obstacle i the poof hee ad also i [MS2] is the use of the esult fom [BS]. Michael Schmuceschlage showed us how to ovecome this obstacle i the poof of [MS2]: Istead of usig [BS] oe ca use Popositio 4. i [ScSc] which says that ay multiple of the ` uit ball has lage o eual Gaussia measue tha the same multiple of the uit ball of ay othe om o IR whose ellipsoid of maximal volume is S ;. This ca eplace the st ieuality o the last lie of p. 542 i [MS2] (with m = ad =. The chage fom the Gaussia measue to the spheical measue is stadad.) It follows that at least fo c= log the aswe to the uestio above is positive. 3
What is the \isomophic' vesio of Dvoetzy's Theoem fo spaces with o-tivial cotype? It is ow that i this case oe has a vesio of Dvoetzy's theoem with a much bette depedece of the dimesio of the Euclidea sectio o the dimesio of the space ([FLM] o see [MS], 9.6). We do ot ow if oe ca exted this theoem i a simila \isomophic" way as the theoem above. The popositio below gives such a extesio ude the additioal assumptio that the space also has o-tivial type. Recall that it is a majo ope poblem whethe a -dimesioal omed space with o-tivial cotype has a subspace of dimesio [=2] which isoftype 2 (with the type 2 costat depedig o the cotype ad the cotype costat oly) o at least of some o-tivial type. If this ope poblem has a positive solutio, the ext popositio would imply the desied \isomophic" cotype case of the theoem. The poof we setch hee (as well as the statemet of the popositio) uses uite a lot of bacgoud mateial (which ca be foud i [MS]) ad is iteded fo expets. Popositio. Fo evey ad evey 2= <=2, ay -dimesioal omed space, X, cotais a -dimesioal subspace, Y, satisfyig d(y `2 ) K=2 = =. Hee K depeds o <, the cotype costat of X ad the om of the Rademache pojectio i L 2 (X) oly. Up to the exact value of the costat ivolved the esult is best possible ad is attaied fo X = `. Setch of poof. We use the otatios of [MS]. We st d a opeato T : `2! X fo which `(T )`(T ; ) K: whee K depeds o the om of the Rademache pojectio i L 2 (X) oly (see [MS], 5.4.). Next we use the \lowe boud theoem" of the st amed autho ([MS], 4.8) to d a 3=4 dimesioal subspace E `2 fo which (T je ) ; C`(T ; )= p fo a absolute costat C. By a theoem of Figiel ad Tomcza ([FT] o [MS], 5.6), thee exists a futhe subspace F E of dimesio lage tha =2 fo which T jf K ;=2 =2;=`(T je ) K ;=`(T ) fo K depedig o ad the cotype costat ofx oly. This educes the poblem to the followig: Give a om o IR =2 fo which C ; x 2 x K =2;= x 2 fo all x, fo costats C K depedig oly o p, ad the type p ad cotype costats of X, ad fo which M = R S =2; x =, d a subspace Y of dimesio as euied i the statemet of the popositio. Sice we have to tae cae of the uppe boud oly, this ca be accomplished by the usual \cocetatio" method as descibed i the st few chaptes of [MS]. 4
The fact that, fo some absolute costat ad fo all, ` does ot have - dimesioal subspaces of distace smalle tha ;=2 =2 = = to `2 follows fom the method developed i [BDGJN] (o see [MS], 5.6). Oe just eed to eplace the costat 2 i 5.6 of [MS] by a geeal costat d ad follow the poof to get a lowe boud o d. Refeeces [BDGJN] G. Beett, L. E. Do, V. Goodma, W. B. Johso, ad C. M. Newma, O ucomplemeted subspaces of L p, <p<2, Isael J. Math. 26 (977), 78-87. [BS] J. Bougai ad S.J. Szae, The Baach - Mazu distace to the cube ad the Dvoetzy- Roges factoizatio, Isael J. Math. 62 (988), 69{8. [FLM] T. Figiel, J. Lidestauss, ad V.D. Milma, The dimesio of almost spheical sectios of covex bodies, Acta. Math. 39 (977), 53{94. [FT] T. Figiel ad. Tomcza-Jaegema, Pojectios oto Hilbetia subspaces of Baach spaces, Isael J. Math. 33 (979), 55{7. [Gl] E. D. Glusi, The octahedo is badly appoximated by adom subspaces, Fuct. Aal. Appl. 2 (986), {6. [Gl2] E. D. Glusi, Extemal popeties of othogoal paallelepipeds ad thei applicatios to the geomety of Baach spaces, Math. USSR Sboi 64 (989), 85{96. [Mi] V. D. Milma, A ew poof of the theoem of A. Dvoetzy o sectios of covex bodies, Fuct. Aal. Appl. 5 (97), 28{37 (taslated fom Russia). [MS] V. D. Milma ad G. Schechtma, Asymptotic theoy of ite dimesioal omed spaces, Lect. Notes i Math. 2 (986), Spige-Velag, Beli. [MS2] V.D. Milma ad G. Schechtma A \isomophic" vesio of Dvoetzy's theoem, C. R. Acad. Sci. Pais t. 32 Seie I (995), 54{544. [Pi] G. Pisie, The volume of covex bodies ad Baach space geomety (989), Cambidge Uiv. Pess. [ScSc] G. Schechtma ad M. Schmuceschlage, A cocetatio ieuality fo hamoic measues o the sphee, Opeato Theoy: Advaces ad Applicatios 77 (995), 255{273. V.D. Milma G. Schechtma Depatmet of Mathematics Depatmet of Theoetical Mathematics Tel Aviv Uivesity The Weizma Istitute of Sciece Ramat Aviv, Isael Rehovot, Isael email: vitali@math.tau.ac.il email: gideo@wisdom.weizma.ac.il 5