L p moments of random vectors via majorizing measures

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1 L p oents of rando vectors via ajorizing easures Olivier Guédon, Mark Rudelson Abstract For a rando vector X in R n, we obtain bounds on the size of a saple, for which the epirical p-th oents of linear functionals are close to the exact ones uniforly on a convex body K R n. We prove an estiate for a general rando vector and apply it to several probles arising in geoetric functional analysis. In particular, we find a short Lewis type decoposition for any finite diensional subspace of L p. We also prove that for an isotropic log-concave rando vector, we only need n p/2 log n saple points so that the epirical p-th oents of the linear functionals are alost isoetrically the sae as the exact ones. We obtain a concentration estiate for the epirical oents. The ain ingredient of the proof is the construction of an appropriate ajorizing easure to bound a certain Gaussian process. 1 Introduction In any probles of geoetric functional analysis it is necessary to approxiate a given rando vector by an epirical saple. More precisely, given a rando vector X R n, we want to find the sallest nuber such that the properties of X can be recovered fro the epirical easure 1/ δ X j, constructed with independent copies X 1,..., X of the vector X. In particular, for p 2 and for y R n, we want to approxiate the oents E X, y p by the epirical averages 1/ X j, y p with high probability. Moreover, we require this approxiation to be unifor over y belonging to soe convex syetric set in R n. A proble of this type was considered Research was supported in part by NSF grant DMS

2 in [5]. Forulated in analytic language, it asks about finding the sallest and a set of points x 1,... x X such that for any function f fro an n-diensional function space F L 1 X, µ), 1 ε) f 1 1 fx j ) 1 ε) f 1. Another exaple of such probles originates in Coputer Science. The probabilistic algorith for estiating the volue of an n-diensional convex body, constructed by Kannan, Lovász, and Sionovits [13] required to bring the body to a nearly isotropic position as a preliinary step. To this end, one has to saple rando points x 1,..., x in the body L so that the epirical isotropy tensor will be close to the exact one, naely 1 x j x j 1 x x dx < ε. 1) voll) This proble was attacked with different probabilistic techniques. The original estiate of [13] was significantly iproved by Bourgain [4]. Using the decoupling ethod he proved that = Cε)n log 3 n vectors x 1,... x uniforly distributed in the body L satisfy 1) with high probability. This estiate was farther iproved to Cn/ε 2 ) log 2 Cn/ε 2 ) in [25], [26]. The proof in [25] used ajorizing easures, while the later proof in [26] was based on the non-coutative Khinchine inequality. These probles were put into a general fraework by Giannopoulos and Milan [9], who related the to the concentration properties of a rando vector. Let α > 0 and let ν be a probability easure on X, Ω). For a function f : X R define the ψ α -nor by f ψα = inf{λ > 0 exp f /λ) α dν 2}. X Chebychev s inequality shows that the functions with bounded ψ α -nor are strongly concentrated, naely ν{x fx) > λt} C exp t α ). Let µ be a Borel easure in R n. It is called isotropic if R n x x dµx) = Id, where Id is the identity operator in R n. Note that this noralization is consistent with the one used in [13, 25, 26]. The noralization used in 2 L

3 [19, 9] differs fro it by the ultiplicative coefficient L 2 µ, where L µ is the isotropic constant of µ see [19]). The paper [9] considers isotropic easures which satisfy the ψ α -condition for scalar products:, y ψα C for all y S n 1. Here and below C, c,... denote absolute constants, whose value ay change at each occurrence. Note that by Borell s lea, any log-concave easure in R n satisfies the ψ 1 -condition [20], [19]. Let p 1 and let µ be an isotropic log-concave easure satisfying the ψ α condition for scalar products with soe α [1, 2]. The central result of [9] provides an estiate for the inial size of a set of independent rando vectors X 1,..., X distributed according to the easure µ such that the epirical p-oents satisfy the inequality ) 1/p 1 Γ 1 p) X, y p Γ 2 p), y S n 1. 2) The ψ α -condition iplies that the L p µ) and L 2 µ)-nors of the function f y x) = x, y are equivalent. Thus the inequality 2) eans that the epirical p-oent of f y is equivalent to the real p-oent up to a constant coefficient. In the present paper we use a different approach to this proble based on the ajorizing easure technique developed by Talagrand [28]. This approach lead to breakthrough results in various probles in probabilistic cobinatorics and analysis see [28] and references therein). In a siilar context the ajorizing easures were applied in [27] to select sall alost orthogonal subatrices of an orthogonal atrix, and in [25] to prove the estiate 1) with sall. To state the results we have to introduce soe notation. Let R n,, ) be a Euclidean space, and let 2 be the associated Euclidean nor. For a syetric convex body K in R n, we denote by K the nor, whose unit ball is K, and by K o = {y R n x K, x, y 1} the polar of K. We assue that the body K has the odulus of convexity of power type q 2 see Section 2 for the definition). Classical exaples of convex bodies satisfying this property are unit balls of finite diensional subspaces of L q [6] or of non-coutative L q -spaces like Schatten trace class atrices [29]). We denote by D the radius of the syetric convex set K i.e. the sallest 3

4 D such that K DB2 n. For every 1 q +, we define q to be the conjugate of q, i.e. 1/q + 1/q = 1. Given a rando vector X in R n, let X 1,..., X be independent copies of X. Let K R n be a convex syetric body. Denote by V p K) = sup 1 X j, y p E X, y p the axial deviation of the epirical p-oent of X fro the exact one. We would like to bound V p K) under inial assuptions on the body K and rando vector X. This will allow us to choose the size of the saple for which this deviation is sall with high probability. Although the resulting stateent is pretty technical, it is applicable to a wide range of probles arising in geoetric functional analysis. We discuss soe exaples in Sections 3, 4. To bound such rando process, we ust have soe control of the rando variable ax 1 j X j 2. To this end we introduce the paraeter κ p, X), which plays a key role below ) 1/p κ p, X) = E ax X j p 2. 1 j We prove the following estiate for V p K). Theore 1 Let K R n,, ) be a syetric convex body of radius D. Assue that K has odulus of convexity of power type q for soe q 2. Let p q and let q be the conjugate of q. Let X be a rando vector in R n, and let X 1,..., X be independent copies of X. Assue that C p,λ log ) 2/q for soe δ < 1. Then D κ p, X)) p δ 2 sup E X, y p EV p K) 2δ sup E X, y p. The constant C p,λ in Theore 1 depends on p and on the paraeter λ in the definition of the odulus of convexity of power type q see Section 2.1 for the definition). 4

5 Note that inial assuptions on the vector X are enough to guarantee that EV p K) becoes sall for large. Indeed, assue that the variable X 2 possesses a finite oent of order p + ε for soe positive ε. Then κ p, X) E X j p+ε 2 ) 1/p+ε 1/p+ε E X p+ε 2 ) 1/p+ε, so the quantity log ) 2/q κ p p,x) tends to 0 when goes to. Moreover, in ost cases, κ p, X) ay be bounded by a sipler quantity: κ p, X) E 1/p X j 2) p e E X s ) 1/s =: em s, 3) where s = axp, log ). Theore 1 iproves the results of [9] in two ways. First, it contains an alost isoetric approxiation of the L p -oents of the rando vector by epirical saples see Theore 2 below). Second, the assuption on the nor of a rando vector X used in Theore 1 is weaker than the ψ α - assuption on the scalar products, appearing in [9]. This allows to handle the situations, where the ψ α -estiate does not hold see e.g. approxiate Lewis decopositions, discussed in Section 3). While Theore 1 cobined with Chebychev s inequality provides a bound for V p K), which holds with high probability, it is often useful to have this probability exponentially close to 1. Using a easure concentration result of Talagrand [15] Theore 6.21), we obtain such probability estiate in Theore 4. We apply Theore 1 to isotropic log-concave rando vectors. This class includes any naturally arising types of rando vectors, in particular a vector uniforly distributed in an isotropic convex body see Section 4 for exact definitions). The epirical oents of log-concave vectors have been extensively studied in the last years [13], [4], [26], [9], [8]. We will prove the following 5

6 Theore 2 For any ε 0, 1) and p 2 there exists n 0 ε, p) such that for any n n 0 ε, p), the following holds: let X be a log-concave isotropic rando vector in R n, let X 1,..., X be independent copies of X, if = C p ε 2 n p/2 log n then for any t > ε, with probabiblity greater than 1 C exp t/c pε ) 1/p ), for any y R n, 1 t)e X, y p 1 X j, y p 1 + t)e X, y p. The constants C p and C p are positive real nubers depending only on p. Theore 2 provides an alost isoetric approxiation of the exact oents, instead of the isoorphic estiates of [9], and achieves it with fewer saple vectors. In the case p = 2, it also iproves the estiate of [26], and extends to the general setting the estiate obtained by Giannopoulos, Hartzoulaki and Tsoloitis [8] for a rando vector uniforly distributed in a 1-unconditional isotropic convex body. The rest of the paper is organized as follows. In Section 2 we forulate and prove the ain results for abstract rando vectors. The key step of the proof of Theore 1 is the estiate of the Gaussian rando process Z y = g j X j, y p, where g j are independent standard Gaussian rando variables N 0, 1). To obtain such estiate we construct an appropriate ajorizing easure and apply the Majorizing easure theore of Talagrand [28]. In Sections 3 and 4, we provide applications of Theore 1. Since we require only the existence of high order oents of the nor of X we can apply Theore 1 to the easures supported by the contact points of a convex body, like in [24], [25], as well as to finding a short Lewis-type decoposition, as described in Section 3. In Section 4, we study in detail the case of log-concave rando vectors X. In the last part of this paper, we extend the results obtained in [9] for a unifor distribution on a discrete cube to a general rando vector X, which satisfies a ψ 2 estiate for the scalar products X, y, y R n. 6

7 2 Maxial deviation of the epirical p-oent 2.1 Stateent of the results Let K R n be a convex syetric body. The odulus of convexity of K is defined for any ε 0, 2) by { } δ K ε) = inf 1 x + y 2, x K = 1, y K = 1, x y K > ε. K We say that K has odulus of convexity of power type q 2 if δ K ε) cε q for every ε 0, 2). It is known see e.g., [23], Proposition 2.4 or [7]) that this property is equivalent to the fact that the inequality x + y 2 q K + λ q x y 2 q K 1 2 x q K + y q K ). holds for all x, y R n. Here λ > 0 is a constant depending only on c and q. Referring to this inequality below, we shall say that K has odulus of convexity of power type q with constant λ. Our ain result is the following theore, which iplies Theore 1 fro the Introduction. Theore 3 Let K R n be a syetric convex body of radius D. Assue that K has odulus of convexity of power type q with constant λ for soe q 2, and let q be the conjugate of q. Let X be a rando vector in R n and let X 1,..., X be independent copies of X. For p q set A = C p λ p log )1/q Dκ p, X)) p/2 and B = sup E X, y p. Then E sup 1 X j, y p E X, y p A 2 + A B. The assuption of Theore 1 reads A 2 δ 2 B, hence A 2 + A B 2δB. Thus, Theore 1 follows iediately fro Theore 3. 7

8 Reark. In fact we shall prove a slightly better inequality. Define κ p,x, K) = E ax 1 j X j 2 2 ) 1/p, ax X j p 2 K 1 j o then Theore 3 holds, if the quantity Dκ p, X)) p/2 is replaced by Dκ p,x, K) p/2. Since K DB n 2, it is clear that κ p,x, K) p/2 D p/2 1 κ p, X) p/2. The proof of this Theore is based on the following lea. Lea 1 Let K R n be a syetric convex body of radius D. Assue that K has odulus of convexity of power type q with constant λ for soe q 2, λ > 0. Let q be the conjugate of q. Then for every p q, and every deterinistic vectors X 1,..., X in R n, E sup ε j X j, y p C p λ p log ) 1/q D ax 1 j X j 2 sup ) 1/2 X j, y 2p 1) where expectation is taken over the Bernoulli rando variables ε j ) 1 j. The proof of the Lea uses a specific construction of a ajorizing easure. It will be presented in part 2.2. Proof of Theore 3. The proof is based on a standard syetrization arguent. We denote by X 1,..., X independent copies of X 1,..., X. Let ε j ) be independent syetric Bernoulli rando variables, which are independent of all others. Then the expectation of V p K) = sup 1 X j, y p E X, y p 8

9 can be estiated as follows: EV p K) = E sup X j, y p E X, y p = E sup Xj, y p E X j, y p) E X E X sup Xj, y p X j, y p) = E X E X E ε sup ε j Xj, y p X j, y p) 2E X E ε sup ε j X j, y p. Therefore, Lea 1 iplies EV p K) C p λ p log )1/q 1 D E X ax X j 2 sup 1 j Since p 2, it is easy to see that ) 1/2 X j, y 2p 1). E X E X 1 ax X j 2 sup 1 j ax X j 2 ax X j p/2 1 K sup 1 j 1 j o ) 1/2 X j, y 2p 1) 1 κ p,x, K) E p/2 1 X sup ) 1/2 X j, y p ) 1/2 X j, y p ) 1/2 κ p,x, K) EV p/2 p K) + sup E X, y p. We get that EV p K) A EV p K) + B) 1/2 where A = C p λ p D log )1/q κ p,x, K) p/2 and B = sup E X, y p which proves the announced result. 9

10 We present now a deviation inequality for the positive rando variable V p K) under the assuption that X 2 satisfies soe ψ α estiate. Mendelson and Pajor [18] studied the sae deviation inequality in the case p = 2 and K = B n 2 using a syetrization arguent. Our approach is based on a concentration result of Talagrand Theore 6.21 in [15]). Theore 4 With the sae notation as in Theore 3, let V p K) be the rando variable V p K) = sup 1 X j, y p E X, y p. Assue that X 2 ψα < for soe 0 < α p. Then there exists a positive constant c α,p depending only on α and p such that where t > 0, PV p K) t) 2 exp t/q) α/p) Q = c α,p EV p K) + ) log )p/α D p X 2 p ψ α. Reark. Observe that in the typical case, Q is of the order EV p K) for which we ay use Theore 3. By Lea 2 see below), therefore, using Theore 3, where κ p, X) Cp log ) 1/α X 2 ψα Q C α,p 2A A 1 B) A 1 = λ p D p/2 log +p/2α )1/q X 2 ψα and B = sup E X, y p. For the proof of this theore, we need an eleentary lea. Lea 2 Let δ > 0 and let Z 1,..., Z be independent copies of a rando variable Z. Then ax Z j ψδ C log 1/δ Z ψδ.,..., 10

11 Proof. Note that for any rando variable Y the inequality Y ψδ A is equivalent to Y r CAr 1/δ for all r > 1. Assue that r < log. Then ) 1/r ax Z j r Z j log ) 1/ log r E Z j log ) r/ log... ) 1/ log E Z j log C log 1/δ Z ψδ. If r > log, then using ax,..., a j ar j) 1/r, we get ax Z j r,..., E Z j r ) 1/r 1/r Z r Cr 1/δ Z ψδ. These two inequalities iply the Lea. Proof of Theore 4. To any vector x R n we associate the function f x defined on K by f x : K R y 1 x, y p E X, y p ). Let f X be the rando vector of L K) associated to X. Now we apply Theore 6.21 of Ledoux Talagrand [15] to f X j where the Xj s are independent copies of X. By definition, and f X L K) 1 sup f Xj L K) = V p K), ) X, y p + sup E X, y p Dp X p 2 + E X p 2). Theore 6.21 of Ledoux-Talagrand [15] states that if α/p 1, there exists a constant c α,p depending only on α/p such that ) V p K) ψα/p c α,p EV p K) + ax f X j L K) ψα/p. 1 j 11

12 Moreover, Lea 2 iplies This proves that ax 1 j f X j L K) ψα/p 2Dp = 2Dp ax 1 j X j p 2 ψα/p ax 1 j X j 2 p ψ α. ax 1 j X j 2 ψα Clog ) 1/α X 2 ψα. 4) ) log )p/α V p K) ψα/p c α,p EV p K) + D p X 2 p ψ α. The deviation inequality follows fro the Chebychev inequality. 2.2 Construction of ajorizing easures Let us recall the assuptions of Lea 1. The abient space is R n equipped with a Euclidean structure and we denote by 2 the nor associated. The syetric convex body K has a odulus of convexity of power type q 2 with a constant λ, which eans that x, y R n, x + y 2 q K + λ q x y 2 and satisfies also the inclusion K DB n 2, which eans that q K x R n, x 2 D x K. 1 2 x q K + y q K ). 5) Let p q 2, and X 1,..., X be fixed vectors in R n. We define the rando process V y for all y R n by V y = ε j X j, y p, where ε j are independent syetric Bernoulli rando variables. It is well known that this process satisfies a sub-gaussian tail estiate: y, y R n, t > 0, ) P V y V y t) 2 exp ct2 d 2 y, y) 12

13 where d 2 y, y) = X j, y p X j, y p ) 2. Instead of working with this function which is not a etric, it will be preferable to consider the following quasi-etric d 2 y, y) = X j, y y 2 X j, y 2p 1) + X j, y 2p 1)). The following propositions state inequalities that we will need to prove Lea 1. Proposition 1 gives soe inforation concerning the geoetry of the balls associated to the etric d and Proposition 2 explains relation between etric d, new Euclidean nor and the following nor defined by x = ax 1 j X j, x. We denote by B ρ x) the ball of center x with radius ρ for the quasi-etric d. Proposition 1 For all y, y K dy, y) 2 y y sup dy, y) p dy, y), 6) ) 1/2 X j, y 2p 1), 7) y y D ax 1 j X j 2 y y K. 8) Moreover, the quasi-etric d satisfies the generalized triangle inequality, and for any point x, the ball B ρ x) is a convex set: for all u 1,..., u N R n and all x, y, z R n, N 1 du 1, u N ) 2p du i, u i+1 ) and d 2 x, y + z 2 ) 1 d 2 x, y) + d 2 x, z) ). 9) 2 i=1 To prove it, we will need the following basic inequalities on real nubers. Lea 3 For every x, y R + and p 2, we have x p y p p x y x 2p 2 + y 2p 2 10) 13

14 Moreover, if fs, t) = s t s 2p 2 + t 2p 2 then for all r 1,..., r N R and for all r, s, t R, N 1 fr 1, r N ) 2p fr i, r i+1 ) i=1 fr, s + t)/2) 2 fr, s) 2 + fr, t) 2) /2 Proof. The first inequality is straightforward. To prove the second one, consider two cases. When r 1 r N 0, since r 1 r N r 1 2p 2 + r N 2p 2 2 r 1 p r N p, the conclusion follows fro the triangle inequality and inequality 10). When r 1 r N 0, we can assue without loss of generality that r 1 0 and r N 0. Then fr 1, r N ) = r 1 + r N ) 2r p 1 + r N p ). r 2p r N 2p 2 r 1 + r N )r p r N p 1 ) Let < N be a nuber such that r 0 and r Then r p 1 + r N p 1 i=1 r i p r i+1 p + r p + r +1 p + Cobining the previous inequalities with 10), we get fr 1, r N ) 2 1 i=1 N 1 i=+1 r i p r i+1 p + 2r p + r +1 p ) + 2 r i p r i+1 p. N 1 i=+1 1 2p fr i, r i+1 ) + 2r r +1 )r p 1 + r +1 p 1 ) + 2p i=1 N 1 i=+1 fr i, r i+1 ) 1 2p fr i, r i+1 ) + 2 2fr, r +1 ) + 2p i=1 14 N 1 i=+1 r i p r i+1 p fr i, r i+1 )

15 which proves the announced result. The last inequality follows fro the fact that for p 2, the function v 1 v) v 2p 2 ) is convex on R, which can be checked by coputing the second derivative. Proof of Proposition 1. Inequalities 6) and 9) clearly follow fro the three inequalities proved in Lea 3. Inequalities 7) and 8) follow fro siple observations about d and the fact that K DB2 n. Proposition 2 Let M = sup X j, y 2p 1). For a fixed u K, we define the Euclidean nor Eu associated to u by z 2 E u = X l, z 2 X l, u 2p 1), z R n. l=1 Then the following inequality holds for all z, z R n : d 2 z, z) 2 4 p 1 z z 2 E u + M z z 2 z u 2p 2 K Proof. By hoogeneity of the stateent, we can assue that M = sup X j, y 2p 1) = 1. + z u 2p 2 K )). For any z R n, let L z = {l {1,..., } Xl, z 2 X l, u }. Then by convexity of the function t t 2p 2, we have X l, z 2p 1) 2 2p 3 X l, z u 2p 1) + 2 2p 3 X l, u 2p 1) l L z l L z l L z 2 2p 3 X l, z u 2p 1) + 1 X l, z 2p 1), 2 l L z l L z which proves since M = 1) that for any z R n, l L z X l, z 2p 1) 4 p 1 z u 2p 2 K. Hence, for any z, z R n, X l, z z 2 X l, z 2p 1) z z 2 X l, z 2p 1) l L z l L z 4 p 1 z u 2p 2 K z z 2. 15

16 For any l / L z we have X l, z 2 X l, u, so l/ L z X l, z z 2 X l, z 2p 1) 4 p 1 X l, z z 2 X l, u 2p 1). The sae inequalities hold if we exchange the roles of z and z. To copute d 2 z i, z j ), we split the su in four parts and apply the inequalities above: d 2 z, z) = l=1 X l, z z 2 X l, z 2p 1) + X l, z z 2 X l, z 2p 1) l=1 = l L z X l, z z 2 X l, z 2p 1) + l/ L z X l, z z 2 X l, z 2p 1) + l L z X l, z z 2 X l, z 2p 1) + l/ L z X l, z z 2 X l, z 2p 1) 2 4 p 1 z z 2 E u + z z 2 z u 2p 2 K + z u 2p 2 K )). Proof of Lea 1. By inequality 6), we ay treat V y as a sub-gaussian process with the quasi-etric p d. By hoogeneity of the stateent, we can assue that sup X j, y 2p 1) = 1. Denote Q = ax 1 j X j 2. We want to show that E sup V y C p λ p Q log ) 1/q D. 11) By Proposition 1, the diaeter of the set K with respect to the etric d is bounded by 2 2QD. Let r be a fixed nuber chosen such that r = cp 2 for a large universal constant c and k 0 be the largest integer such that r k 0 2 2QD. The proof of inequality 11) is based on the ajorizing easure theory of Talagrand [28]. The following theore is a cobination of Proposition 2.3, Theore 4.1 and Proposition 4.5 of [28]. Note that assuing that r 2, one can set Kr) = C in Proposition 2.3, and K2, 1, r) = C in Proposition 4.5. Theore [28] Let r 2. Let φ k : K R + for k k 0 be a faily of aps satisfying the following assuption: there exists A > 0 such that for any point x K, for any k k 0 and any N N 16

17 for any points x 1,..., x N B r kx) with dx i, x j ) r k 1, i j H) we have ax φ k+2x i ) φ k x) + 1 i=1,...,n A r k log N. Then for any fixed y 0 K, E sup V y V y0 c A sup φ k x). k k 0,x K To obtain the conclusion of Lea 1, set y 0 = 0. To coplete the proof, we have to define the functionals φ k : K R +. Let k 1 be the sallest integer such that r k 1 QD/ n. For k k 1 + 1, set φ k x) = log r + n Q D log ) 1/q k l=k 1 r l log1 + 4QDrl ). Note that in this range of k the functionals φ k do not depend on x. We shall show that with this choice of φ k, the condition H) follows fro the classical voluetric estiate of the covering nubers. For k 0 k k 1, the functionals φ k are defined by φ k x) = in{ y q K, y B 4pr kx)} + k k 0 log. Since q 2 then 1 q 2 and log ) 1/q log. It is easy to see using definitions of k 0 and k 1 that sup x K,k k 0 φ k x) c. We shall prove that our functionals satisfy condition H) for A = Cλ) p Q D log ) 1/q where C is a large nuerical constant. That will conclude the proof of Lea 1 with a new constant C. Proof of condition H). Let N N, x K, x 1,..., x N B r kx) with dx i, x j ) r k 1. We have to prove that ax φ k+2x i ) φ k x) i=1,...,n 17 r k log N Cλ) p Q D log ) 1/q.

18 For k k 1 1, we always have φ k+2 x i ) φ k x) n log1 + 4QDr k+2 ) QDlog ) 1/q r k 2. Since the points x 1,..., x N are well separated in the etric d, they are also well separated in the nor K. Indeed, by 7) and 8), we have x i x j K r k 1 /QD 2. By the classical voluetric estiate, the axial cardinality of a t-net in a convex syetric body K R n with respect to K does not exceed 1 + 2/t) n. Therefore, log N n log QDr k+1 ), which proves the desired inequality. The case k 0 k k 1 2 is uch ore difficult. Our proof uses estiates of the covering nubers, in particular, the dual Sudakov inequality [22]. Recall that the covering nuber NW, X, t) is the inial cardinality of X -balls of radius t needed to cover the W. For j = 1,..., N denote by z j K the points which satisfy z j q K = in{ y q K, y B 4pr k 2x j)}. Denote by u K a point such that u q K = in{ y q K, y B 4pr kx)}. Set θ = ax z j q K j u q K. Then we have ax j φ k+2 x j ) φ k x) = θ + 2. We shall prove that log θ + 2 log r k log N/A. 12) Since dx i, x j ) r k 1, z l B 4pr k 2x l ), and d satisfies a generalized triangle inequality, the points z j ) 1 j N reain well separated. Indeed, r k 1 dx i, x j ) 2pdx i, z i )+dz i, z j )+dz j, x j )) 2pdz i, z j )+16p 2 r k 2 and since r = cp 2, we have dz i, z j ) r k 1 /cp 18

19 for all i j. Recall that r = cp 2. Using again the generalized triangle inequality, we get that dx, z j ) 2pdx, x j ) + dx j, z j )) 2pr k + 4pr k 2 ) 4pr k. It eans that z j B 4pr kx), u B 4pr kx), and the convexity of the balls for the quasi-etric d proved in Proposition 1 iplies u + z j )/2 B 4pr kx). Since K has odulus of convexity of power type q, inequality 5) holds. By the definition of u, we get that for all j = 1,..., N λ q z j u 2 q K 1 2 z j q K + u q K ) z j + u 2 q K z j q K u q K 2 θ 2. This proves that j = 1,..., N, z j u K 2λθ 1/q. Let δ > 0. Consider the set U = u + 2λθ 1/q K which contains all the z j s and let S be the axial nuber of points in U that are 2δ separated in. Then U is covered by S subsets of diaeter saller than 2δ in etric, and so S NU,, 2δ). Set δ = c p λ 1 p r k θ 1/q 1 where the constant c will be chosen later. Since U = u + 2λθ 1/q K and K DB2 n,the dual Sudakov inequality [22] iplies log S log NB2 n,, δ/dλθ 1/q ) c D λ θ 1/q E G / δ. Here G denotes a standard Gaussian vector in R n. It is well known that E G = E ax,..., X j, G c Q log. We consider now two cases. First, assue that S N. Then by previous estiate and the definition of δ, we get log N c Q λ D log θ 1/q / δ θ c p r k Q λ p D log which easily proves 12) since q 2). 19

20 The second case is when S N. Since U is covered by S balls of diaeter saller than 2δ in, there exists a subset J of {1,..., N} with #J N such that i, j J, z i z j 2δ. By Proposition 2 applied to the Euclidean nor defined by we get that y 2 E u = X l, y 2 X l, u 2p 1), l=1 d 2 z i, z j ) 2 4 p 1 z i z j 2 E u + 4 p λ 2p 2 θ 2p 2)/q δ 2). Since θ 1 and q p, the definition of δ iplies 4 2p λ 2p 2 θ 2p 2)/q δ 2 4 c) 2p r 2k θ 2p/q 1) 4 c) 2p r 2k. Recall that dz i, z j ) r k 1 /cp and r = cp 2. Hence, r 2k /cp 6 dz i, z j ) p 1 z i z j 2 E u + 24 c) 2p r 2k. Choosing c sall enough, we get that for all i, j J, z i z j Eu r k 1 c p. Since K DB n 2, we have the following estiate for the covering nubers: #J NU, Eu, c p r k 1 ) = NK, Eu, c p r k 1 /2λθ 1/q ) NB2 n, Eu, c p r k 1 /2λθ 1/q D). Recall that G denotes a standard Gaussian vector in R n. By the dual Sudakov inequality [22], we have log NB n 2, Eu, cp r k 1 2λDθ 1/q ) Cp r k+1 θ 1/q λ D E G Eu C p r k+1 θ 1/q λ D E G 2 E u ) 1/2. Since for all y K, X j, y 2p 1) 1, we obtain E G 2 E u = X l 2 2 X l, u 2p 1) Q 2. l=1 20

21 Since #J N, we have log N C p r k+1 λdqθ 1/q with a universal constant C. Moreover, by Young s inequality θ 1/q log ) 1/q θ/q + 1/q log )) and since q 2 q and λ 1, we get log N Cλ) p r k+1 D Q log ) θ 1/q + 2 ). log This copletes the proof of 12) and the proof of condition H) for the functionals φ k. 3 Approxiate Lewis decoposition It is well known that if E is an n-diensional subspace of L p, then E is 1 + ε)-isoorphic to an n-diensional subspace of l N p with N depending on n, p and ε. Lewis [16] proved that any linear subspace E of l N p possesses a special decoposition of the identity. More precisely, there exists a Euclidean structure on E with the scalar product,, vectors y 1,..., y N E and scalars c 1,..., c N > 0 such that i, y i, y i = 1, N ) 1/p x E = c i x, y i p, x E, Id E = i=1 N c i y i y i. i=1 Denote by H, H ) the linear space E equipped with this Euclidean structure. Recall that p denotes the conjugate of p. In the following Theore, we prove that both spaces E and H can be 1 + ε)-ebedded in l p and l 2 respectively via the sae linear operator T : R N R, whenever is of the order of ε 2 n p/2 log 2/p n/ε 4/p ). This extends a classical result of Bourgain, Lindenstrauss and Milan [5] and [15] for a better dependance on ε) and soe results in [24] concerning the nuber of contact points of a convex body needed to approxiate the identity decoposition. 21

22 Theore 5 Let E be an n-diensional subspace of L p for soe p 2. Then for every ε > 0 there exists a Euclidean structure H = E,, ) on E and points x 1,..., x in E with Cp n ) ε 2 np/2 log 2/p Cp n ) ε 4/p ε 2 np/2 log 2 ε 4/p such that j, x j H = 1 and for all y E, n ) 1/p 1 ε) y E y, x j p 1 + ε) y E n ) 1/2 1 ε) y H y, x j ε) y H. Proof. Let X be the rando vector taking values y i with probability c i /n. Then for all y E, E X, y p = y p E /n and E X, y 2 = y 2 H/n and X H = 1. We will apply Theore 1 twice: first tie for the unit ball of E, and then for the unit ball of H. Since E is a subspace of L p, by Clarkson s inequality [6], B E has odulus of convexity of power type p with constant λ = 1. Fro Lewis decoposition, we get y E y H n p y E which eans that for D = n p, B H B E DB H. Let X 1,..., X be independent copies of X, then sup E X, y p = 1/n and κ p, X) = y B E E ax 1 j X j p H ) 1/p = 1. Applying Theore 1 with δ = ε, we get that if C p n p/2 log ) 2/p /ε 2, then E sup n X j, y p y p E y B E ε. Now, we apply Theore 1 for K = B H which clearly has odulus of convexity of power type 2 i.e. satisfies inequality 5) for q = 2). In that case, D = 1, and ) 1/2 sup E X, y 2 = 1/n and κ 2, X) = E ax X j 2 2 = 1. y H 1 1 j 22

23 Applying Theore 1 for q = p = 2 and δ = ε, we get that if C 2 n log /ε 2, E sup n X j, y 2 y 2 H ε. y H 1 Choosing the sallest integer such that, for a new constant C, C p ε 2 np/2 log n/ε 4/p ) 2/p we get by Chebychev s inequality that there exist vectors x 1,..., x of Euclidean nor 1 such that for all y E, n x j, y p y p E ε y p E and n x j, y 2 y 2 H ε y 2 H which gives the desired result. 4 Isotropic log-concave vectors in R n We investigate the case of X being an isotropic log-concave vector in R n or also a vector uniforly distributed in an isotropic convex body). Let us recall soe definitions and classical facts about log-concave easures. A probability easure µ on R n is said to be log-concave if for every copact sets A, B, and every λ [0, 1], µλa + 1 λ)b) µa) λ µb) 1 λ. There is always a Euclidean structure, on R n for which this easure is isotropic, i.e. for every y R n, E X, y 2 = x, y 2 dµx) = y 2 2. R n A particular case of a log-concave probability easure is the noralized unifor Lebesgue) easure on a convex body. Borell s inequality [3] see also [20, 19]) iplies that the linear functionals x x, y satisfy Khintchine 23

24 type inequalities with respect to log-concave probability easures. Naely, if p 2, then for every y R n, E X, y 2 ) 1/2 E X, y p ) 1/p Cp E X, y 2) 1/2, 13) or in other words, y ψ1 C E X, y 2) 1/2. We have stated in 3) that it is easy to deduce soe inforation about the paraeter κ p, X) fro the behavior of the oent M s of order s = axp, log ) of the Euclidean nor of the rando vector X. These oents were studied for a rando vector uniforly distributed in an isotropic 1- unconditional convex body in [2], and for a vector uniforly distributed in the unit ball of a Schatten trace class in [12], where it was proved that when s c n, M s is of the sae order as M 2 up to constant not depending on s). Very recently, Paouris [21] proved that the sae stateent is valid for any log-concave isotropic rando vector in R n. We state precisely his result. Theore [21] There exist constants c, C > 0 such that for any log-concave isotropic rando vector X in R n, for any p c n, E X p 2) 1/p C E X 2 2) 1/2. Fro this sharp estiate, we will deduce the following Lea 4 Let X be an isotropic log-concave rando vector in R n and let X j ) 1 j be independent copies of X. If e c n, then for any p 2 ) 1/p { κ p, X) = E ax X j p C n if p log 2 1 j C p n if p log Proof. Since X is isotropic, and for every y R n, E X, y 2 = y 2 2, we get E X 2 2 = n. By Borell s inequality [3], q 2, E X q 2) 1/q Cq n. Therefore if p log, ) 1/p E ax X j p 2 E 1/p X j 2) p Cp 1/p n Cp n. 1 j 1 j If p log, by 3) ) 1/p E ax X j p 2 e 1 j 24 E X log 2 ) 1/ log.

25 Since e c n, log c n, the Theore of Paouris iplies ) 1/ log C n, E X log 2 which concludes the proof of the Lea. Corollary 1 Let X be an isotropic log-concave rando vector in R n, and let X j ) 1 j be independent copies of X. Then for every e c n ax 1 j X j 2 ψ1 C n. Proof. By Lea 4, we know that ) 1/r r 2, E ax X j r 2 Cr n 1 j which proves the claied estiate for the ψ 1 -nor. Reark. Recall that for a rando isotropic log-concave vector, Borell s inequality iplies that X 2 ψ1 C n. Therefore, a direct application of Lea 2 is not enough to obtain the desired estiate. We are now able to give a proof of Theore 2. It is based on the estiates of κ p, X) proved above. Proof of Theore 2. Let ε 0, 1) and p 2 and set n 0 ε, p) = c p + ε 4/p where c p depends only on p. For any n n 0 ε, p), for any log-concave isotropic rando vector X in R n, set V p = sup 1 X j, y p E X, y p y B2 n where X 1,..., X are independent copies of X. Assue that p log and e c n then by Lea 4, we know that κ p, X) p c p 1n p/2. We shall use Theore 1 with K = B2 n which is uniforly convex of power type 2 with constant 1 and for which D = 1. By 13), 1 sup E X, y p p p, y B2 n 25

26 therefore Theore 1 iplies that for every δ 0, 1), satisfying C p n p/2 log ) δ 2, we have E sup 1 X j, y p E X, y p 2δp p. y B2 n By taking δ such that 2δp p = ε, we deduce that if C p ε 2 n p/2 lognε 4/p ) then EV p ε. Since n n 0 ε, p), it is easy to see that if = C p ε 2 n p/2 log n, then C p ε 2 n p/2 lognε 4/p ), e c n and p log which allows us to use the estiate EV p ε. To get a deviation inequality for V p, we will apply a result siilar to Theore 4. We know by Corollary 1 that ax 1 j X 2 ψ1 C n. Following the proof of Theore 4 and replacing inequality 4) by the previous estiate, we easily see that ) V p ψ1/p C p EV p + 2np/2. Since C p ε 2 n p/2 log n = then EV p ε and 2n p/2 / ε and we deduce fro the Chebychev inequality that for any t > 0, PV p t) C exp t/c pε ) 1/p ). Therefore, for any t ε, with probability greater than 1 C exp t/c pε ) 1/p ), V p t which eans that y R n, 1 X j, y p E X, y p t y p 2. 26

27 Since y 2 = E X, y 2 ) 1/2 E X, y p ) 1/p, we get the claied result of Theore 2. Reark. Since by Borell s inequality 13), for any y R n, y 2 = E X, y 2) 1/2 E X, y p ) 1/p Cp E X, y 2) 1/2 = Cp y 2, it is clear that Theore 2 iproves the results of Giannopoulos and Milan [9]. 5 When the linear functionals associated to the rando vector X satisfy a ψ 2 condition Let start this section considering the case when X is a Gaussian vector in R n. Let X j, j = 1,...,, be independent copies of X. For t R denote by X t,y the Gaussian rando variable X t,y = t j X j, y. Observe that if p denotes the conjugate of p, then ) 1/p sup X t,y = X j, y p. t Bp Let Z and Y be Gaussian vectors in R and R n respectively. Using Gordon s inequalities [10], it is easy to show that whenever E Z p ε 1 E Y 2 i.e. for a universal constant c, c p p p/2 ε p n p/2 ) ) 1/p E Z p E Y 2 E inf X j, y p y S n 1 E sup y S n 1 ) 1/p X j, y p E Z p + E Y 2, where E Z p + E Y 2 )/E Z p E Y 2 ) 1 + ε)/1 ε). It is therefore possible to get with high probability with respect to the diension n, see [11]) a faily of rando vectors X 1,..., X such that for every y R n, A y 2 1 X j, y p ) 1/p A 1 + ε 1 ε y 2. 27

28 This arguent significantly iproves the bound on in Theore 2 for Gaussian rando vectors. In this part we will be interested in isoorphic oent estiates instead of alost isoetric as in Theore 2). We will be able to extend the estiate for the Gaussian rando vector to rando vector X satisfying the ψ 2 condition for linear functionals y X, y with the sae dependance on. Recall that a rando variable Z satisfies the ψ 2 condition if and only if for any λ R E expλz) 2 expcλ 2 Z 2 2 ). We prove the following Theore 6 Let X be an isotropic rando vector in R n such that all functionals y X, y satisfy the ψ 2 condition. Let X 1,..., X be independent copies of X. Then for every p 2 and every n p/2 1 E sup y B2 n ) 1/p X j, y p c p. Note that the results of Part 3 of [9] follow iediately fro Theore 6, since the rando vector with independent ±1 coordinates satisfies the ψ 2 condition for scalar products. Proof. Since X is isotropic, X, y ψ2 c X, y 2 = c y. Hence, for any λ R Writing E exp λ X, y 2e cλ2 y 2 2. = X t,y X t,y = tj t j) X j, y + t j X j, y y ), it is easy to find a new constant c 1 such that for every t, t B p, y, y B n 2 and every λ R +, E expλ ) 2e cλ2 t t y y 2 2 ). 28

29 This eans that ψ2 c t t y y 2 2) 1/2, and so X t,y is a sub- Gaussian rando process with respect to the distance d t, y); t, y ) ) = t t y y 2 2) 1/2. Let G t,y = Z, t + Y, y, where Z R and Y R n are two independent Gaussian vectors. Then E Gt,y G t,y 2) 1/2 = d t, y); t, y ) ) The natural etric for the rando process X t,y is bounded by the etric of the process G t,y. The Majorizing Measure theore of Talagrand [28] iplies that E sup t,y) V X t,y C sup t,y) V G t,y for any copact set V R R n. Therefore, 1 E sup y B2 n ) 1/p X j, y p = 1 E sup 1/p C E sup 1/p t B p C p + n 1/p ). sup G t,y = y B2 n This proves that if n p/2, then 1 E sup y B2 n t B p sup y B n 2 C ) E Z p + E Y 1/p 2 ) 1/p X j, y p c p, t j X j, y as claied. Reark. Let X be an isotropic rando vector in R n satisfying the ψ 2 estiate for the scalar products. It is not difficult to see, using Corollary 2.7 in [9], that if Cn, then with probability greater than 3/4 c 2 y 2 1 ) 1/p X j, y p 29

30 for every y R n. Therefore, using Theore 6, it is easy to deduce that if n p/2, then with probability greater than 1/2 y R n c 2 y 2 1 ) 1/p X j, y p c 1 p y 2 with universal constants c 1, c 2 1. This generalizes results of [9] and gives an isoorphic version of the result of Klartag and Mendelson [14] valid for every p 2. Acknowledgeent. A part of this work was done when the first author was visiting University of Missouri Colubia). We wish to thank this institution for its hospitality. References [1] S. Alesker, ψ 2 -estiate for the Euclidean nor on a convex body in isotropic position, Geoetric aspects of functional analysis Israel, ), 1 4, Birkhäuser, Basel, [2] S. Bobkov, S. Nazarov, On convex bodies and log-concave probability easures with unconditional basis, Geoetric aspects of functional analysis, 53 69, Lecture Notes in Math., 1807, Springer, Berlin, [3] C. Borell, Copleents of Lyapunov s inequality, Math. Ann ), [4] J. Bourgain, Rando points in isotropic convex sets, Convex geoetric analysis Berkeley, CA, 1996), 53 58, Cabridge Univ. Press, Cabridge, 1999 [5] J. Bourgain, J. Lindenstrauss and V. Milan, Approxiation of zonoids by zonotopes, Acta Math ), no. 1-2, [6] J. A. Clarkson, Uniforly convex spaces, Trans. Aer. Math. Soc ), no. 3, [7] T. Figiel, An exaple of infinite diensional reflexive Banach space nonisoorphic to its Cartesian square, Studia Math ), [8] A. A. Giannopoulos, M. Hartzoulaki and A. Tsoloitis, Rando points in isotropic unconditional convex bodies, J. Lond. Math. Soc ), [9] A. A. Giannopoulos and V. D. Milan, Concentration property on probability spaces, Adv. Math., ), no. 1, [10] Y. Gordon, Gaussian processes and alost spherical sections of convex bodies, Ann. Probab ), no. 1,

31 [11] Y. Gordon, On Milan s inequality and rando subspaces which escape through a esh in R n Geoetric aspects of functional analysis 1986/87), , Lecture Notes in Math., 1317, Springer, Berlin, 1988 [12] O. Guédon, G. Paouris, Concentration of ass on the Schatten classes, to appear in Annales de l Institut Henri Poincaré, Probabilités et Statistiques. [13] R. Kannan, L. Lovász, M. Sionovits, Rando walks and an O n 5 ) volue algorith for convex bodies, Rando Structures Algoriths ), no. 1, [14] B. Klartag, S. Mendelson, Epirical processes and rando projections, to appear in J. Funct. Anal. [15] M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, Berlin, [16] D. R. Lewis, Finite diensional subspaces of L p, Studia Math ), no. 2, [17] L. Lovász and M. Sionovits, Rando walks in a convex body and an iproved volue algorith, Rando Structures Algoriths ), no. 4, [18] S. Mendelson and A. Pajor, On singular values of atrices with independent rows, to appear in Bernoulli. [19] V. D. Milan and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a nored n-diensional space. Geoetric aspects of functional analysis ), , Lecture Notes in Math., 1376, Springer, Berlin, [20] V. D. Milan and G. Schechtan, Asyptotic theory of finite-diensional nored spaces, Springer, Berlin, 1986 [21] G. Paouris, Concentration of ass on syetric convex bodies, to appear in Geo. Funct. Anal. [22] A. Pajor and N. Toczak-Jaegerann, Subspaces of sall codiension of finitediensional Banach spaces, Proc. Aer. Math. Soc ), no. 4, [23] G. Pisier, Martingales with values in uniforly convex spaces, Israel J. Math ), no. 3-4, [24] M. Rudelson, Contact points of convex bodies, Israel J. Math ), [25] M. Rudelson, Rando vectors in isotropipc position, MSRI preprint. [26] M. Rudelson, Rando vectors in isotropipc position, J. Funct. Anal ), no. 1, [27] M. Rudelson, Alost orthogonal subatrices of an orthogonal atrix, Israel J. of Math ), [28] M. Talagrand, Majorizing easures: the generic chaining, Ann. Probab ), no. 3,

32 [29] N. Toczak-Jaegerann, The oduli of soothness and convexity and the Radeacher averages of trace classes S p 1 p < ) Studia Math ), Olivier Guédon: Université Paris 6, Institut de Mathéatiques de Jussieu, Projet Analyse Fonctionnelle, 4, place Jussieu, Paris, France; e-ail: guedon@ccr.jussieu.fr Mark Rudelson: Departent of Matheatics, University of Missouri, Colubia, MO 65211, USA; e-ail: rudelson@ath.issouri.edu 32