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The Limitations of the Method

arXiv:math/9305203v1 [math.FA] 11 May 1993Random Banach spaces.The Limitations of the MethodPiotr Mankiewicz ∗Institute of MathematicsPolish Academy of SciencesWarsawStanis law J. Szarek †Case Western Reserve UniversityClevelandWe shall study the properties of typical n–dimensional subspaces of lN∞= (IR N, ∥·∥∞), orequivalently, typical n–dimensional quotients of lN1 = (IR N, ∥· ∥1), where the meaning whatis typical and what is not is defined in terms of the Haar measure µn,N on the Grassmannmanifold Gn,N of all n–dimensional subspaces of IR N.In [Gl.2], Gluskin proved that a “typical” n–dimensional subspace E of ln2∞enjoys theproperty∥P∥≥ck√n log n,for every projection P : E →E, with min{rank P, rank (Id −P)} = k, where c is a numeri-cal constant. In particular, if k ≥nα, α > 12, then no projection P on E with both rank P andcorank P greater than k can be “well” bounded.

Several other results,[Sz.1],[Sz.2],[Ma.1],[Ma.2]showed that a “typical” proportional (i.e. dim E ≈βN for some “fixed” β ∈(0, 1)) subspaceE of lN∞has the property that every “well” bounded operator on E is indeed a “small” per-turbation of a multiple of the identity λIdE.

However, the estimates on the distance betweenT and λIdE have been done in terms of the geometry of IR N rather than E itself. In thisnote, we obtain the estimates on the distance between T and λIdE in intrinsic terms of thegeometry of E, namely, in terms of the Gelfand numbers of T −λIdE (Sections 2 and 3).On the other hand, we show in Section 4, that if k ≤n1/2 then a “typical” n–dimensionalsubspace E of lN∞(for any N ≥n) contains a k–dimensional well-complemented subspaceG isomorphic to lkp with either p = 2 or p = ∞and therefore admits operators which are“fairly” far away from the line {λIdE}λ∈IR .We shall employ the standard notation of local theory of Banach spaces as used in e.g.[F-L-M].

For basics on multivariate Gaussian random variables the reader is referreded to[T].∗Part of this research was done while this author has been visiting the Case Western Reserve University.Supported in part by a grant from KBN (Poland).†Supported in part by a grant from the National Science Foundation (U.S.A.).1

1Generic subspaces. Equivalent Gaussian approachLet Pn for n ∈IN be a sequence of properties of n–dimensional Banach spaces and letf : IN →IN be an increasing function.We shall say that P = {Pn}n∈IN is a genericproperty of n–dimensional subspaces of lN∞where N = f(n) ifffor every n ∈IN we haveµn,N{E ∈Gn,N | E satisfies Pn} ≥1 −εn,for some ε ∈(0, 1), ε independent of n.In the sequel, we shall say that a generic n–dimensional subspace of lN∞has a property P = {Pn}n∈IN rather then that the propertyP is a generic one.

Since the dual of an n–dimensional subspace E of lN∞is the quotientF = lN1 /E⊥, where lN1 = (IR N, ∥· ∥) and E⊥is the orthogonal complement of E in IR N, thenotions of generic properties of quotients of lN1 and generic subspaces of lN1 can be defined.E.g.a generic quotient of lN1satisfies a property P iffthe corresponding (via duality)geometric generic subspace of lN∞satisfies the dual property P ∗.In the context of quotients it is more convenient to consider an equivalent approach.Let g be an IR n–valued Gaussian vector distributed according to the N(0, n−1IdIR n) law(i.e., the covariance matrix of g is n−1IdIR n, or the coordinates of g are i.i.d. N(0, 1/n)random variables).

Let g1, g2, . .

. , gN be independent copies of g and Γ– an n × N matrixwhose columns are g1, g2, .

. .

, gN; alternatively, Γ = Γ(ω) can be described as a Gaussiann × N matrix with i.i.d. N(0, 1/n) entries.

If we think of Γ as of a (random) linear mapfrom IR N to IR n, lN1 /ker Γ is a random quotient of lN1 . By the rotational invariance of theGaussian measure, this model is “measure theoretically” equivalent to the one describedat the begining of this section and based on the Haar measure on the Grassmannian.

Stillequivalently, we may consider the random norm on IR n, whose unit ball is a random absoluteconvex bodyB = B(ω) = absconv {gj : j = 1, 2, . .

. , N} = Γ(BN1 ) ,where BN1 is the unit ball of lN1 .In the sequel we will need some basic facts about Gaussian vectors, Gaussian matricesand “Gaussian bodies”.The first lemma is an elementary consequence of the formulae forGaussian density in IR d.Lemma 1.1 If g is a Gaussian random variable with distributionN(0, d−1IdIR d), then(i) IE ∥g ∥22 = 1(ii) P (∥g ∥2 ≥λ) ≤exp (−dλ2/8)for λ ≥2(iii) P (∥g ∥2 ≤t) ≤te1/2d(iv) P12 ≤∥g ∥2 ≤2≥1 −exp (−c0d) ,where c0 is a universal constant.The next lemma can be derived from the first one using a standard ǫ–net argument (cf.

[Sz.3], Lemma 2.8). Another, more precise argument (giving e.g.

c > 14 and C < 2) can befound in [Si].2

Lemma 1.2 Let k ≤12N and Λ be an N ×k matrix with all i.i.d. Gaussian N(0, 1) entries.ThenP( c∥x ∥2 ≤k−1/2∥Λx ∥2 ≤C∥x ∥2 for every x ∈IR k) ≥1 −exp (−c1d) ,where c, C, c1 are universal constants.We have an immediateCorollary 1.3 If k and N are as in Lemma 1.2, g1, g2, .

. .

, gN are i.i.d. Gaussian randomvariables with distribution N(0, k−1IdIR k) and B = B(ω) = absconv {g1, g2, .

. .

, gN}, thenP(B ⊃ck−1/2D) ≥1 −exp (−c1d) ,where D stands for the Euclidean unit ball in IR k.The last lemma gives more precise information about the random bodies B(ω) ⊂IR k (cf. [Gl.3],[Gl.4]).Lemma 1.4 If g1, g2, .

. .

, gN, B are as in Corollary 1.3 and 2k ≥N ≥2k, then(i) PB ⊃c′qlog N/kkD≥1 −exp (−c2k)(ii) P(vol B /vol D )1/k ≤C′qlog N/kk≥1 −exp (−c2k) ,where c, C′, c2 > 0 are universal constants.RemarkThe volume estimate from Lemma 1.4 (ii) actually holds for any B =absconv {x1, x2, . .

. , xN} as long as we controlmax {∥xj ∥2 : j = 1, 2, .

. .

, N},which in our case we do by Lemma 1.1 (iv) (see [Ca-P] or [Gl.4]).2The proportional case.Our starting point is the following result ([Ma.2], Proposition 2.3).Theorem 2.1 There is a numerical constant c > 0 such that for every n ≥2 there is anorm ∥· ∥Xn on IR n such that(i) Xn = (IR n, ∥· ∥) is isometrically isomorphic to a quotient of l2n1 ,(ii) ∥x ∥2 ≤∥x ∥Xn ≤∥x ∥1for every x ∈IR n,(iii) for every T ∈L(IR n) there are λT ∈IR , VT ∈L(IR n) and a linear subspaceET ⊂IR n with dim ET > 7n8 , such thata) VT = T + λIdIR n ,b) | λT | ≤c∥T ∥Xn ,c) ∥VT |ET ∥≤cn−12∥T ∥Xn .3

In fact, Theorem 2.1 holds for “sort of” generic n-dimensional quotients of l2n1 (cf. [Ma.1]).However in order to adapt the above result to our present setting, we need to make a coupleof observations.

First, the condition (ii), which is not crucial for our purposes, has to besuperseded by the properties listed in Corollary 1.3 and Lemma 1.4 (the condition of type(ii) may be, moreover, achieved, up to a universal constant and restricted to the span of, say,the first n/2 unit vectors e1, e2, . .

. , en/2; see [Ma.-T]).

Next (and more importantly), we haveto point out that in all constructions leading to Theorem 2.1–like statements ([G1.1], [G1.2],[Ma.1], [Ma.2], [Ma.-T], [Sz.1], [Sz.2] etc. ), “generic” had a somewhat different (and slightlyless natural) meaning.

We take here an opportunity to present a remark which rectifies thisproblem. What happens is that when ensuring the condition (iii) from Theorem 2.1, we needto work with the setT = {T ∈L(IR n) : ∥T∥Xn ≤1}or, more specifically, with nets of T in the lN1 metric.

Now we have (hs(·) is the Hilbert-Schmidt norm)If X, Y are quotients of lN1 , endowed with the canonical inner product and T : X →Yverifies ∥T : X →Y ∥≤1, then hs(T) ≤N1/2.The above statement is shown by estimating hs(T) = hs(T ∗) = π2(T ∗) by the π2-norm of(a restriction of) the formal identity Id : lN∞→lN2 , which is N1/2. If N = 2n, it can be shownin a standard way that T ′ = {T ∈L(IR n) : hs(T) ≤N1/2} admits a δ-net in the ∥·∥ln2 metricwhich is of cardinality not exceeding (C/δ)n2, where C is a numerical constant, and this canbe easily incorporated into existing proofs of Theorem 2.1-like statements.

Unfortunately,we do not see how to handle in the same “unified” way e.g. the casse considered in Theorem3.1.We now prove the following.Theorem 2.2 There is a numerical constant K > 0 such that if Xn is the quotient spacefrom Theorem 2.1 then, for every operator T ∈L(IR n), we haveinf{c n2 (T −λIdIR n) | λ ∈IR } ≤Kn−12∥T∥Xn.We recall here that, for an operator u : X →Y , the kth Gelfand number of u is defined byck(u) = inf{∥u|Z∥: Z ⊂X, codim Z < k}.Because of a well-known duality relation between the Gelfand numbers and the so-called“Kolmogorov numbers” dk(·) (namely dk(u∗) = ck(u)), one can also state our results interms of the latter ones.

Note a slight abuse of notation; in the above and in what followswe pretend that n2 and similar expressions are integers.Observe that Theorem 2.2 is the best possible. This follows either from Corollary 4.3below or from the fact that, for an n-dimensional “generic” quotient of lN1 , (for any N > n),and for a “generic” element of O(n), the left hand side of the inequality in Theorem 2.2 is oforder n while the right hand side–at most of order n1/2.

In fact, any n-dimensional normedspace can be represented on IR n so that the last remark holds.4

Proof of Theorem 2.2 Obviously, it is enough to prove the theorem for every operatorT ∈L(IR n) satisfyinginf{c n2 (T −λIdIR n) |λ ∈IR } = 1. (2.1)To this end, fix such an operator T. It is well known, [Gl.1], [Sz.2], thatvol (BXn) ≤c1nnandBXn ⊃1√nB2n.Hence, by [Sz.-T], we infer that there exists an 3n4 - dimensional subspace, say E, of IR n suchthatBXn ∩E ⊂c2√nB2E.(2.2)Claim.

For every λ ∈IR n the operatorTλ |E = (T −λIdIR n) |E(2.3)has at least n/4 s - numbers greater than or equal to 1/c2 .Indeed, if this was not the case then, by (2.2), we would have for some λ0 ∈IR1c2> ∥Tλ0 |E0 : (E0, n−12∥. ∥2 ) →(IR n, n−12∥.

∥2) ∥≥1c2∥Tλ0 |E0 : (E0, ∥. ∥Xn ) →(IR n, ∥.

∥Xn) ∥,(2.4)where E0 is an n2 - dimensional subspace such that ∥Tλ0 |E0 ∥2 < 1/c2. But (2.4) implies thatc n2 (T −λ0IdIR n) < 1,a contradiction with (2.1), which concludes the proof of the claim.In particular, the Claim yields that for every λ ∈IR the operator T −λIdIR n has at leastn/4 s - numbers greater then or equal to 1/c2, which means that for every subspace F ⊂IR nwith dim F ≥7n8 we have∥T −λIdIR n |F ∥2 ≥1c2.Hence, by Theorem 2.1 (iii), c), we infer that1 ≤c2cn−12∥T ∥Xnwhich implies ∥T ∥Xn ≥Kn12, where K = (cc2)−1.

This concludes the proof of the theorem.✷5

3The l1 - type estimates.For a Banach space X = (IR n, ∥. ∥B) we setM∗B =ZSn−1 ∥x ∥∗Bdµ(x),where dµ stands for the normalized Lebesque measure on the unit sphere Sn−1 and ∥.

∥∗Bdenotes the dual norm to ∥. ∥B; this is the “average width” of B.In the sequell we shall need the following fact which can be found in [P–T], Theorem 1(cf.

[Pi.2 ], Theorem 1.3).Fact I There exist a numeric constant C > 1 such that for every symmetric convex bodyB ⊂IR n and for every k = 1, 2, . .

. , n−1 there exists a subspace Ek ⊂IR n with codim Ek = ksuch thatB ∩Ek ⊂CM∗Brnk D ∩Ek.Recall that for an operator T ∈L(IR n) we say that T ∈Mn(α, β), where α, β > 0 iffthere is a linear subspace F ⊂IR n with dim F ≥α such that∥PF ⊥Tx∥2 ≥β∥x∥2 for every x ∈F,(where PF ⊥denotes the orthogonal projection onto F ⊥).

Also, for γ > 0, we denote˜Mn(γ) =n/2[k=1Mn(k, γ/k).The following fact has been proved in [Sz.2]Fact II A generic n - dimensional quotient Xn of ln21enjoys the property∥T∥Xn ≥c1γ√n log nfor every T ∈˜Mn(γ).From Fact I and Fact II we deduceTheorem 3.1 A generic n - dimensional quotient Xn of ln21has the property that for everyT ∈L(IR n) we haveinf {nXi=1ci(T −λIdIR n) | λ ∈IR } ≤cn2/3qlog3 n ∥T∥Xn,where c is a numerical constant.6

Remark. It is imaginable that one could strengthen the above inequality to get O(n1/2)∥T∥Xnon the right hand side; ◦(n1/2)∥T∥Xn is impossible, see the comments following Theorem 2.2.Proof In order to simplify the notation we shall assume that n = 2k for some k ∈N.

LetXn be a generic n–dimensional quotient of l1n2. Obviously, it suffices to prove thatnXi=1ci(T) ≤cn2/3qlog3 n ∥T∥Xn,(3.5)for every T ∈L(IR n) satisfying tr T = 0.

To this end, fix T ∈L(IR n) such that tr T = 0 andnXi=1ci(T) = n.(3.6)Then, there is i ≤k such that2i−1c2i ≥nlog n.(3.7)If 2i ≤n2/3 then we have∥T∥Xn ≥c2i ≥2n1/3log n ,which combined with (3.6), by a standard homogenuity argument, yields (3.5) and we aredone.Thus, assume that 2i > n2/3. It is well known (cf.

Corollary 1.3) that n−1/2D ⊂BX(E)and that M∗X(E) ≤c2√n−1 log n, (see e.g. [F-L-M]).

Thus, applying Fact I we infer that thereexists a linear subspace F2i−1 ⊂IR n with codim F2i−1 = 2i−1 such that1√nB2n ∩F2i−1 ⊂BXn ∩F2i−1 ⊂c2Cslog n2i−1 B2n ∩F2i−1.(3.8)Claim. The operator T |F2i−1 has at least 2i−1 s - numbers greater than or equal to(c2C)−1sn2i−1 log3 n .Indeed, assume to the contrary that there exists a linear subspace F ⊂F2i−1 withcodim F = 2i such that∥T |F : (F, ∥· ∥2) →(IR n, ∥· ∥2 ) ∥< (c2C)−1sn2i−1 log3 n .

(3.9)Then, by (3.8) and (3.7) we have(c2C)−1sn2i−1 log3 n > ∥T |F : (F, n−1/2∥· ∥2) →(IR n, n−1/2∥· ∥2) ∥=(c2C)−1s2i−1n log n∥T |F : (F, c2Cslog n2i−1 ∥· ∥2) →(IR n, n−1/2∥· ∥2) ∥≥7

(c2C)−1s2i−1n log n∥T |F : (F, ∥· ∥Xn) →(IR n, ∥· ∥Xn) ∥≥(3.10)(c2C)−1s2i−1n log nc2i(T) ≥(c2C)−1s2i−1n log nn2i−1 log n =(c2C)−1sn2i−1 log3 n ,a contradiction which concludes the proof of the claim.Now, observe that if the median s - number of Tsn/2(T) < 12(c2C)−1sn2i−1 log nthen, by [Ma.1], Lemma 2.6, we obtain thatT ∈˜Mn 132(c2C)−1s2inlog n!,while ifsn/2(T) ≥12(c2C)−1sn2i−1 log nthen, by [Ma.1], Theorem 3.1, we infer thatT ∈˜Mnc32 (c2C)−1sn32i−1,where c3 < 1 is the constant from Theorem 3.1 in [Ma.1]. In the first case Fact II yields∥T∥Xn ≥c132(c2C)−1s2ilog n ≥c132(c2C)−1vuut n2/3log n ,while in the second case, by Fact II, we get∥T∥Xn ≥c1c32 (c2C)−1snlog3 n .thus, by (3.6), in both cases we obtainnXi=1ci(T) ≤32c2C(c1c3)−1n2/3qlog3 n ∥T∥Xn ,which proves (3.5) with c = 32c2C(c1c3)−1 and completes the proof of the theorem.✷8

4The positive statements.In this section we prove several “positive” statements about existence of “nontrivial” oper-ators on generic Banach spaces, which will show that the results of the preceeding sectionare “essentially” optimal (cf. Cor.

4.3). Results similar to some of the presented below wereobtained independently by Gluskin [Gl.5].

The first of these statements will also show that,for generic Banach spaces, the following conjecture due to Pisier [Pi.1] (and usually referredto as the “dichotomy conjecture”) holds.There exist C > 1 and a function f : IR + →IR + with limλ→∞f(λ) = ∞, such that ifE ⊂lN∞, then there exists F ⊂E, dim F = k ≥f(dim E/ log N) verifying d(lk∞, F) ≤C.At the time of this writing the conjecture is open for general E. However, we haveProposition 4.1 There exist universal constants C, c > 0 such that if E is a generic quo-tient of lN1 , dim E = d (resp. a generic subspace of lN∞), then there is a subspace F ⊂E,dim F = k ≥c min{d12, d/ log N} verifying(i) d(lk1, F) ≤C (resp.

d(lk∞, F) ≤C)(ii) F is C–complemented in E.Proposition 4.2 If, in the notation of Proposition 4.1, N ≥d2, then there is G ⊂E,dim G = h ≥min{c log N, d}, satisfying(i)’ d(lh2, G) ≤C(ii)’ G is C–complemented in E.RemarkIf, in Proposition 4.2, one assumes that N > d1+α for α ∈(0, 1), one gets G whichis C/√α complemented in E; for arbitrary N and d we get Cq log Nlog N/d – complementation.Corollary 4.3 A “generic” d–dimensional quotient of lN1 contains a C–complemented sub-space C–isomorphic to lkp, k ≥c√d, either for p = 1 or p = 2 (resp. a “generic” subspace oflN∞contains such a subspace with either p = ∞or p = 2).Proof of Corollary 4.3If log N < d12, we use Prop.4.1 to get a C–complementedsubsapce C–isomorphic to lk1 (resp.

lk∞). If log N > d12 (hence N > d2), use Prop 4.2 to geta C-complemented Hilbertian subspace.✷Proof of Proposition 4.1We will prove the statement for a random quotient Q; the“subspace” variant follows by duality.Recall that gj = Qej, j = 1, 2, .

. .

, N are independent Gaussian vectors with distributionN(0, d−1IdIR d) and that the unit ball B of our random quotient is absconv{g1, g2, . .

. , gN}.Clearly, we can assume that N ≤exp{cd}.

It then follows from Lemma 1.1 (iv) thatP(12 ≤∥gj ∥≤2 for j = 1, 2, . .

. , N) ≥1 −exp(−c1d)(4.11)9

provided that c is chosen to satisfy c ≤c0/2, where c0 is the constant from Lemma 1.1.Moreover, if k ≤d/2 and A ⊂{1, 2, . .

. , N} with |A| = k, then (cf.

Lemma 1.2)∥Xj∈Atjgj ∥≥c2Xj∈A|tj|212for all choices of scalars (tj)j∈A(4.12)with the similar probability as in (4.11). In fact, sinceNk< ( Nek )k, (4.12) happens for allsuch A with the same estimate on the probability as in (4.11) provided k log Nk ≤c3d.

Inparticular, this happens if k ≤cdlog N (we do not use this fact here).In the next step we shall show that, for fixed A ⊂{1, 2, . .

., N} with |A| ≤k =c min {d12,dlog N }, and E = [gj | j ∈A], we haveP∥PEgj ∥≤k12 | j /∈A≥1 −exp −c4dk! (4.13)Observe that (4.12) and (4.13) imply the conclusion of the Proposition 4.1 with C = c−12 .To this end, note that the operator u : lk1 →E sending {e1, e2, .

. .

, ek} into {gj | j ∈A} is ofnorm 1, while ∥u−1PE ∥≤c−12(notice that (4.12) implies that ∥· ∥B ≤c−12 k12∥· ∥2 on E).To prove (4.13), assume for simplicity that A = {1, 2, . .

. , k}.

For fixed {g1, g2, . .

. , gk},and hence fixed E, ˜gj = PEgj, j = k + 1, k + 2, .

. .

, N are independent E–valued Gaussianvectors with N(0, 1dIdE) distribution. In particular, E∥˜gj∥22 = kd and, by Lemma 1.1 (ii),P∥˜gj∥≥k−12≤exp−k8" k−1/2(k/d)1/2#2= exp −d8k!,(we used the fact that k ≤cd12, and hence k−12 ≥c−1( kd)12 ≥2( kd)12).

Note a slight abuse ofnotation; in fact the expectation and the probability above are conditional on {gj | j ≤k}To deduce (4.13), we need to know that N exp−d8kis small. This happens e.g.

whenk ≤d16 log N and can be forced by the proper choice of c.✷Proof of Proposition 4.2Clearly, it is enough to prove the Proposition for quotients oflN1 . The variant for subspaces of lN∞will follow by duality.As follows from Lemma 1.4 (i), the unit ball B of a generic d–dimensional quotient of lN1contains a Euclidean ball Dr with radius r = c′qlog (N/d)d. On the other hand, by Lemma 1.4(ii), |B||Dr|!

1d≤C′′,and hence the so–called volume ratio of B with respect to Dr remainds bounded by auniversal numerical constant. In particular, (cf.

[Sz-T]), this implies that, say, for k ≤d/2,and for a generic k–dimensional subspace G and some universal constant C,G ∩Dr ⊂G ∩B ⊂C(G ∩Dr),10

which means that G considered as a subspace of B is C–Euclidean.To conclude the proof we will show that if P is the orthogonal projection onto the generick–dimensional subspace G of IR N and k ≥log N, thenP(B) ⊂C1skd(G ∩D),(4.14)where D denotes the Euclidean unit ball in IR n. This will suffice, since if k ≃log N andif C is large enough then C1qkd ≤Cc′qlog N/dd(remember that N ≥d2 and thereforelog N ≤2 log Nd ). Thus, (4.14) implies P(B) ⊂CDr.

To prove (4.14) observe that for afixed G (and hence fixed P) and for j ∈{1, 2, . .

. , N}, Pgj is a Gaussian random vector withdistribution N(0,qkdIdG).

Therefore, by Lemma 1.1 (ii),P∥Pgj∥2 ≤λskd≥1 −exp (−18λ2k)for λ ≥2. Choosing λ sufficiently large and using the fact that k ≥log N we can obtaina similar estimate for P∥Pgj∥2 ≤λqkd | j = 1, 2, .

. .

, N. Finally, by the rotational invari-ance of the joint distribution of gj’s and the Fubbini theorem we deduce that a “generic” B,a “generic” P and G satisfy (4.14).✷11

References[Ca.–P]Carl, B. & Pajor A. Gelfand numbers of operators with values in a Hilbertspace.

Invent. Math.

94 (1988), 479–504. [Gl.1]Gluskin, E. D., The diameter of the Minkowski compactum is roughly equal ton.

Func. Anal.

i Priloz., 15 (1) (1981), 72–73 (in Russian). [Gl.2]Gluskin, E. D., Finite dimensional analog of a space without basis.

Dokl. ANSSSR 261 (5) (1981), 1046–1050 (in Russian)[Gl.3]Gluskin, E. D., Oktaeder is poorly approximable by random subspaces.

Func.Anal. i Priloz., 20 (1) (1986), 14–20 (in Russian)[Gl.4]Gluskin, E. D., Extremal properties of orthogonal parallelograms and theirapplications to the geometry of Banach spaces.

Mat. Sbor.

136 (178) (1988), 85–96 (in Russian)[Gl.5]Gluskin, E. D., Personal communications. [F-L-M]Figiel, T., Lindenstrauss, J., & Milman, V. D., The dimension of almostspherical sections of convex bodies.

Acta Math. 139 (1977), 56–94.

[Ma.1]Mankiewicz, P., Factoring the identity operator on a subspace of l∞n . StudiaMath., 95 (1989), 134–139.

[Ma.2]Mankiewicz, P., Subspace mixing properties of operators in IR n with applica-tions to Gluskin spaces. Studia Math., 88 (1988), 51–67.

[Ma.–T]Mankiewicz, P. & Tomczak–Jaegermann, N., Random subspaces and quo-tients of finite dimensional Banach spaces. Preprint.Odense University (1998).

[P–T]Pajor, A. & Tomczak–Jaegermann, N, Subspaces of small codimensions offinite dimensional Banach spaces.

Proc.AMS, 97 (1986), 637-642. [Pi.1]Pisier, G., Remarques sur un resultat non publie de B. Maurey.

Seminaired’Analyse Fonctionnelle, Ecole Polytechnique, (1980–1981) exp. 5.

[Pi.2]Pisier, G., A new approach to several results of V. Milman. J. Reine Angew.Math., 393 (1989), 115–131.[Si.

]Silverstein, J., The smallest eigenvalue of a large dimensional Wishart matrix.Ann. Probab., 13 (1985), 1364–1368.

[Sz.1]Szarek, S. J., On finite dimensional basis problem with an appendix on nets ofGrassmann manifolds. Acta Math.

151 (1983), 153–179. [Sz.2]Szarek, S. J., On the existence and uniqueness of complex structure and spaceswith few operators.

Trans. Amer.

Math. Soc., 293 (1986), 339–353.12

[Sz.3]Szarek, S. J., Spaces with large distance to ln∞and random matrices. Amer.

J.Math. 112 (1990), 899–942.

[Sz-T]Szarek, S. J. & Tomczak-Jaegermann, N., On nearly Euclidean decompo-sition for some classes of Banach spaces.

Comp. Math., 40 (1980), 367–385.

[T]Tong, Y., Probability Inequalities in Multivariate distributions. Academic Press,New York.13

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