Houston Journal of Mathematics. c 2002 University of Houston Volume 28, No. 3, 2002

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1 Houston Journal of Mathematics c 00 University of Houston Volume 8, No. 3, 00 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES N.J. KALTON AND A.E. LITVAK Communicated by Gilles Pisier Abstract. We study the existence of cubic quotients of finite-dimensional quasi-normed spaces, that is, quotients well isomorphic to l k for some k. We give two results of this nature. The first guarantees a proportional dimensional cubic quotient when the envelope is cubic; the second gives an estimate for the size of a cubic quotient in terms of a measure of nonconvexity of the quasi-norm.. Introduction It is by now well-established that many of the core results in the local theory of Banach spaces extend to quasi-normed spaces (cf. [], [3], [4], [7], [8], [9], [0], [3], [5], [6], [7] for example). In this note we give two results on the local theory of quasi-normed spaces which are of interest only in the non-convex situation. Let us introduce some notation. Let X be a real finite-dimensional vector space. Then a p-norm on X, p (0, ], is a map x x (X R) so that: (i) x > 0 if and only if x 0. (ii) αx = α x for α R and x X. (iii) x + x p x p + x p for x, x X. Then (X, ) is called a p-normed space. For the purposes of this paper a quasi-normed space is always assumed to be a p-normed space for some p (0, ] (note that by Aoki-Rolewicz theorem on quasi-normed space one can introduce an 99 Mathematics Subject Classification. Primary: 46B07, 46A6. Key words and phrases. Quasi-normed spaces, quotients. The first author was supported by NSF grant DMS and the second author was supported by a Lady Davis Fellowship. 585

2 586 N.J. KALTON AND A.E. LITVAK equivalent p-norm ([], [4], [])). The set B X = {x : x } is the unit ball of X. The closed convex hull of B X, denoted by ˆB X, is the unit ball of a norm ˆX on X; the corresponding normed space, ˆX, is called the Banach envelope of X. The set B is called p-convex if for every x, y B and every positive λ, µ satisfying λ p + µ p = one has λx + µy B. Clearly, the unit ball of p-normed space is a p-convex set and, vice versa, a closed centrally-symmetric p-convex set is the unit ball of some p-norm provided that it is bounded and 0 belongs to its interior. If X and Y are p-normed spaces (for some p) then the Banach-Mazur distance d(x, Y ) is defined as inf{ T T }, where the infimum is taken over all linear isomorphisms T : X Y. We let d BX = d X = d(x, l dim X ) and δ BX = δ X = d(x, ˆX). It is clear that δ X is measure of non-convexity; in fact δ X = inf{d(x, Y ) : Y is a Banach space}. We now describe our main results. In Section 3 we investigate quasi-normed spaces X such that ˆX satisfies an estimate d( ˆX, l dim X ) C. It has been known for some time that non-trivial examples of this phenomenon exist []. In geometrical terms this means that the convex hull of the unit ball of X is close to a cube. We show using combinatorial results of Alesker, Szarek and Talagrand [], [0] based on the Sauer-Shelah Theorem [8], [9] that X then has a proportional dimensional quotient E satisfying an estimate d(e, l dim E ) C. A much more precise statement is given in Theorem 3.4. We then use this result in Section 4 to prove that a p-normed space X has a quotient E with dim E c p ln δ X /(ln ln δ X ) and d(e, l dim E ) C p where 0 < c p, C p < are constants depending on p only. Again a more precise statement is given in Theorem 4.. In developing these results, we found it helpful to use the notion of a geometric hull of a subset of R n. Thus instead of considering a p-convex set B X we consider an arbitrary compact spanning set S and then compare the absolutely convex hull S with certain subsets Γ θ S which can be obtained from S by geometrically converging series. Precisely x Γ θ S, θ (0, ), if and only if x = ( θ) n=0 θn λ n s n where s n S and λ n. Note that Γ α α θ Γ θ for every 0 < α < θ <. Our results can be stated in terms of estimates for the speed of convergence of Γ θ S to S as θ. In this way we can derive results which are independent of 0 < p < and then obtain results about p-normed spaces as simple Corollaries. We develop the idea of the geometric hull in Section and illustrate it by restating the quotient form of Dvoretzky s theorem in this language.

3 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES 587. Approximation of convex sets Let S be a subset of R n. Denote by S the absolutely convex hull of S and by S the star-shaped hull of S, i.e. S = {λx : λ, x S}. For each m N we define m S to be the set of all vectors of the form m (λ x + + λ m x m ) where λ k and x k S for k m. If 0 < θ < we define the θ-geometric hull of S, Γ θ S to be the set of all vectors of the form ( θ) k=0 λ kx k where λ k θ k and x k S for k = 0,,. Lemma.. Let S be a p-convex closed set where 0 < p <. Then for 0 < θ < we have ( Γ θ S p /p ( θ) /p) S. Proof. This follows easily from: which in turns from the estimate θ ( θ p ) /p p /p ( θ) /p θ p p( θ). Lemma.. If 3 < θ < and m N then Γ θ m S θ 3θ Γ θ m S. Proof. Note that Hence Now observe This completes the proof. m S m m θ m Γ θ m S θ m( θ m ) θ m = k=0 θ k m S. θ m( θ m ) θ m Γ θ m S. θ m(θ m ) θ ln θ 3 θ.

4 588 N.J. KALTON AND A.E. LITVAK In this section, we make a few simple observations on the geometric hulls Γ θ S. Let us suppose that S is compact and spanning so that S coincides with the unit ball B X of a Banach space X, X. Given q [, ] let T q = T q (X) denote the equal-norm type q constant, i.e. the smallest constant satisfying N Ave ɛ ɛ k =± k x k T q N /q max x k k N k= X for every N. Given an integer N let b N denote the least constant so that N inf ɛ ɛ k =± k x k b N N max x k. k N k= X Given a set A by A we denote the cardinality of A. The following Lemma abstracts the idea of [7], Lemma. Lemma.3. Suppose 3 k= b k m θ. Then < θ <, and let m = m(s) be an integer such that S θ (3θ )( θ) Γ θ m S. Proof. Suppose N N and suppose u N S. Then u = N (x + + x N ) where x k S. Hence there is a choice of signs ɛ k = ± with {ɛ k = } N and N ɛ k x k Nb N. k= X Let v = N ( ɛ k = x k). Then u v X b N. Hence N S N S + b N S. Iterating we get k ms k m S + b j m S which leads to which implies j= S m S + θ S S ( θ) θ Γ θ m S (3θ )( θ) Γ θ m S.

5 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES 589 Proposition.4. (i) Suppose < q and q be such that /q + /q =. Then for ( ) q θ = /q 4 T q we have S Γ θ S. (ii) There exists constant C < so that if m is the largest integer such that X has a subspace Y of dimension m with d(y, l m ) then S 8Γ θ S for θ = (Cm) C log log(cm). Remark. We conjecture that the sharp estimate in (ii) is θ = c/m. Proof. (i) Observe that b N T q N q. Hence b k N T q N q ( q ). k= Let N be the largest integer so that the right-hand side is at most. Applying Lemma.3 with θ 0 = / we obtain The result follows, since ( N /q T q S 4Γ /N S. ) q N and Γ α α θ Γ θ for α < θ. In (ii) we note first by a result of Elton [5] (see also [] for a sharper version) there exist universal constants / c 0 < and C so that b N0 < c 0 for some N 0 Cm. Recall simple properties of the numbers b k. Clearly, for every k, l one has b kl b k b l and (k + l)b k+l kb k + lb l. Thus if b k c 0 < then b l c = ( + c 0 )/ < for every k l k. Therefore we may suppose that N 0 is a power of two, say N 0 = q, q, and b N0 c <. Since b l for every l, we get b N s 0 l c s for every integers s, l 0. Then, taking N = N0 r for some r we have rq b k N = b rq+jrq+l rq c jr rqc r / k= j=0 l= j= provided r c ln q with appropriate absolute constant c.

6 590 N.J. KALTON AND A.E. LITVAK Now take r to be smallest integer larger than c ln q = c ln log N 0. Then by Lemma.3 we obtain S 4Γ /N S for N (C m) C log log(c m) and the result follows. Corollary.5. There are absolute constants c, C > 0 so that if X is a p-normed space then there exists a subspace Y in the envelope ˆX such that dimension of Y is { } ln A m cp exp, ln ln A where A = C(δ X ) p/( p), and d (Y, l m ). Proof. Let S = B X and let m be as in Proposition.4. Then by the proposition we have B X 8Γ θ B X with Thus by Lemma. we obtain θ = (Cm) C log log(cm). B X 8p /p +/p (Cm) ( /p)c log log(cm) B X, i.e. That implies the result. δ X (C m/p) ( /p)c log log(cm). Let us conclude this section with a very simple form of Dvoretzky s theorem recast in this language: Theorem.6. Let η < /3. There is an absolute constant c > 0 so that if S is a compact spanning subset of R n then there is a projection P of rank at least cη log n such that d Γθ P S + η θ for every 3η θ <. Remark. Let ɛ 6/7. Setting θ = 3η = ɛ/ we observe that there is an absolute constant c > 0 so that if S is a compact spanning subset of R n then there is a projection P of rank at least cɛ 4 log n such that d Γɛ/ P S + ɛ.

7 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES 59 Remark. The quotient form of Dvoretzky s theorem for quasi-normed spaces is essentially known and follows very easily from results in [7] (see e.g. [8] for the details). Proof. By the sharp form of Dvoretzky s Theorem (Theorem.9 in [6]) there is a projection P of rank at least cη log n so that d (P S) + η. Let Y = P R n and introduce an inner-product norm on Y so that E (P S) ( + η)e where E = {y : (y, y) }. If y E with y = there exists u P S ( P S) with (y, u). Since u + η we obtain y u (η + η ) / 3η. Hence which implies, for any θ 3η, Hence which proves the theorem. E P S ( P S) + 3η E ( θ)e Γ θ P S ( + η)e. d Γθ P S + η θ, 3. Approximating the cube Let n be an integer. By [n] we denote the set {,..., n}. The n-dimensional cube we denote by B = Bn. D n denotes the extreme points of the cube, i.e. the set {, } n. Given a set σ [n] by P σ we denote the coordinate projection of R n onto R σ, and we denote Bσ := P σ Bn, D σ := P σ D n. As above A denotes the cardinality of a set A. As usual and denote the norm in l and l correspondingly. Theorem 3.. There are constants c > 0 and 0 < C < so that for every ɛ > 0, if S D n with S n( cɛ) then there is a subset σ of [n] with σ ( ɛ)n so that D σ Cɛ P σ ( N S) for some N Cɛ. Proof. We will follow Alesker s argument in [], which is itself a refinement of Szarek-Talagrand [0]. Alesker shows that for a suitable choice of c, if ɛ = s then one can find an increasing sequence of subsets (σ k ) s k=0 so that P σ 0 (S) = D σ0, σ s ( ɛ)n and if τ k = σ k \ σ k for k =,,..., s then there exists α D n so that P τk (S P σ k (P σk α)) = D τk. It follows that if a D τk there exists x S with P σk (x) = 0 and P τk (x) = a.

8 59 N.J. KALTON AND A.E. LITVAK We now argue by induction that D σk a k P σk bk S where a k = k+ and b k = k a k = 4 k k. This clearly holds if k = 0. Assume it is true for k = j, where j s. Then if a D σj we can observe that there exists x a j bj S with P σj x = P σj a. Clearly, P τj x a j bj D τj. Hence there exists x a j bj S with P σj x = 0 and P τj x = P τj x. Finally pick x 3 S so that P σj (x 3 ) = 0 and P τj (x 3 ) = P τj a. Then P σj (x + x + x 3 ) = a and x + x + x 3 a j bj S + a j bj S + S a j ( 4bj + j) b 4bj + j S = a j bj S. j This establishes the induction. We finally conclude that D σs ( s+ )P σs 4 ss and this gives the result, as the case of general ɛ follows easily. Remark. Slightly changing the proof one can show that D σ Cɛ α P σ ( N S) for N Cɛ α, where α = log 3. Lemma 3.. There exist absolute constants c, C > 0 with the following property. Suppose 0 < ɛ < and 0 < k < n are natural numbers with k/n cɛ( ln ɛ). Let S be a subset of R n so that if a D n there exists x S with {i : x i = a i } k. Then there is a subset σ of [n] with σ ( ɛ)n and D σ Cɛ N P σ S for some N Cɛ. Proof. Suppose 0 < k < n and k/n = tɛ( ln ɛ). We shall show that if t is small enough we obtain the conclusion of the lemma. First pick a map a σ(a) from D n [n] so that for each a, σ(a) = k and there exists x S with x i = a i for i σ(a). Then, by a simple counting argument we have the existence of τ [n] so that τ = k and if T = {α D τ : a D n, σ(a) = τ, P τ a = α} then T n n k( n k ). We can estimate (n ) ( n ) ( ) k n k ( ) n k n ne. k k n k n k

9 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES 593 Hence for t / we have ( ) ( ) n ntɛ e ln(e/ɛ) log k ln ( ln ɛ) ln 3ktɛ ( ln t). tɛ It follows that T k( C tɛ), where C t = 3t ( ln t). Choosing t such that C t c/, where c is the constant from Theorem 3., and applying this theorem, we obtain the existence of σ τ, σ ( ɛ/)k ( ɛ)n, with desired property. Theorem 3.3. There are absolute constants c, C > 0 such that if ɛ > 0 and S is a subset of R n with B S db then there is a subset σ of [n] with σ n( ɛ) such that (C/ɛ) Γ θ P σ S for θ = cd ɛ 5 ( ln ɛ). B σ Proof. Let δ = c ɛ and m be the smallest integer greater than c d ɛ 3 ( ln ɛ), where c, c will be chosen later. Suppose first that a D n. Then we can find N N, N m, and x,..., x N S ( S) so that a N (x + + x N ) nd m. Let Ω be the space of all m-subsets of [N] and let µ be normalized counting (probability) measure on Ω. If (ξ i ) N i= denote the indicator functions ξ(ω) = if i ω and 0 otherwise then E(ξ i ) = E(ξi ) = m N, E(ξ m(m ) iξ j ) = N(N ) if i j. Thus and E(ξ i E(ξ i )) = m N m N m(m ) E((ξ i E(ξ i ))(ξ j E(ξ j )) = N(N ) m N if i j. Let y = N (x + + x N ) so that y = E( N m i= ξ ix i ). Then working in the l -norm we have E m N ξ i x i y = i= N m mn(n ) N x i i= N m mn (N ) N x i. i=

10 594 N.J. KALTON AND A.E. LITVAK Hence Since y a nd m E m we have E m N ξ i x i y nd m. i= N ξ i x i a 4 nd m. i= We now suppose that for each ω Ω we have {j : N m i= ξ ix i (j) a(j) > δ} > 4d n/(mδ ). Then we get an immediate contradiction. We conclude that for each a D n there exists x a m S such that x a (j) a(j) δ for at least n( c c ɛ( log ɛ) ) choices of j. Let y a (j) = a(j) if x a (j) a(j) δ and y a (j) = x a (j) otherwise so that y a x a δ. Now suppose c is chosen as a function of c so that we can apply Lemma 3. to obtain the existence of a set σ [n] with σ n( ɛ) and so that D σ Cɛ P σ N {y a : a D n } where C is an absolute constant, and N Cɛ. Then D σ Cɛ P σ Nm S + Cɛ δb σ. Recall that Cɛ δ = Cc so that if we choose c such that Cc = 4 we have D σ K + 4 B σ where K := Cɛ P σ Nm S. Now suppose x Bσ. Let a, a D σ be defined by a (j) = if x(j) and a (j) = otherwise, while a (j) = if x(j) and a (j) = otherwise. Then x (a + a ). Thus B σ K B σ = Cɛ P σ Nm S B σ. This implies for θ = 3 4, B σ 4Cɛ Γ θ P σ Nm S Letting ϕ = θ /(Nm) and applying Lemma. we obtain Γ θ Nm S 6 5 Γ ϕs.

11 QUOTIENTS OF FINITE-DIMENSIONAL QUASI-NORMED SPACES 595 Note that ( 3 4 )/(Nm) (Nm) ln(4/3) cd ɛ 5 ( ln ɛ) for some c > 0 so that the result follows. Theorem 3.4. There is an absolute C > 0 such that if ɛ > 0 and X is a p-normed quasi-banach space with dim X = n and d( ˆX, l n ) d then X has a quotient Y with dim Y n( ɛ) and d(y, l dim Y ) Cp p ɛ 4 5 p ( ln ɛ) p d p. Remark. In [] examples are constructed of finite-dimensional p-normed spaces X n (with 0 < p < fixed) so that d( ˆX n, l dim X n ) is uniformly bounded but lim n δ Xn =. Proof. We can assume B B ˆX db. Then by Theorem 3.3 we can find σ with σ n( ɛ) so that cɛbσ Γ θ P σ B X where θ = cd ɛ 5 ( ln ɛ). Let Y be the space of dimension σ with unit ball B Y = P σ B X. Since B Y is p-convex we have (Lemma.) Γ θ B Y p p (cd ɛ 5 ( log ɛ) ) p BY. Finally observe that for a suitable c > 0: The result then follows. cp p d p ɛ 5 p 4 ( log ɛ) p B σ B Y db σ. 4. Cubic quotients We start this section with the following lemma, which is in fact a corollary of Theorem 3.3. Lemma 4.. Let S be a compact spanning subset of R n and X be the Banach space with unit ball B X = S. Let m be the largest integer such that X has a subspace Y of dimension m with d(y, l m ). Then for every integer k satisfying k m there exists a rank k projection π, so that for some cube Q one has Q Γ b πs CQ, where 0 < b < is an absolute constant. Proof. Let Y be a subspace of X of dimension m so that d(y, l m ). Then if k m there is a linear operator T : Y l k with T and T (B Y ) B k. T can then be extended to a norm-one operator on X and so X has a quotient Z of dimension k so that d(z, l k ). It follows immediately from Theorem 3.3 with ɛ = that there is a further quotient W of Z with dim W k

12 596 N.J. KALTON AND A.E. LITVAK and for some cube Q 0 in W, and fixed constants 0 < b < and < C <, we have Q 0 Γ b π W S CQ 0 where π W is the quotient map onto W. Theorem 4.. There is an absolute constant c > 0 so that if X is a finitedimensional p-normed space, then X has a quotient E with d(e, l dim E ) (cp) /p and dim E c ln A/(ln ln A), where A = (p /p δ X /4) p/( p) (assuming that δ X is large enough). Remark. Take X = l n p so that δ X = n +/p. Then if X has a quotient E of dimension k with d(e, l k ) C ( p then ) ˆX = l n also has such a quotient which implies k cc p ln n = cc p ln. We conjecture that this estimate is δ p/( p) X optimal up to an absolute constant, i.e. that every p-normed space has a cubical quotient of such dimension. As one can see from the proof below we could obtain such an estimate (up to constant depending on p only) if we were able to prove the inclusion with θ = c(m ln m) in Proposition.4. Proof. Let S = B X and m be the largest integer such that X has a subspace Y of dimension m with d(y, l m ). Assume first m k. By Proposition.4 (and its proof) we have B X 4Γ θ B X for θ = /N k, where N k = (Ck) C ln ln(ck). Then, by Lemma., we obtain B X 4p /p (N k ) +/p which implies δ X 4p /p (N k ) +/p. Therefore N k A := (p /p δ X /4) p/( p). Finally we obtain k C ln A/(ln ln A) (of course we may assume that A > e ). Suppose now k C ln A/(ln ln A). By above we have m k. So Lemma 4. implies the existence of absolute constants b, C and a rank k projection π such that Q Γ b πb X C Q for some cube Q. By Lemma. we obtain so that we have (if E = X/π (0)), Γ b πb X p /p ( b) /p πb X d(e, l k ) C p /p ( b) /p. This implies the theorem. Acknowledgment. The work on this paper was started during the visit of the second named author to University of Missouri, Columbia.

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14 598 N.J. KALTON AND A.E. LITVAK Received November 30, 000 Revised version received January 30, 00 (N.J. Kalton) Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65 address: (A.E. Litvak) Department of Mathematics, Technion, Haifa, Israel, address: Current address: Department of Math. and Stat. Sc., University of Alberta, Edmonton, AB, Canada address: