A note on norms of signed sums of vectors

Transcription

1 A ote o orms of siged sums of vectors Giorgos Chasapis ad Nikos Skarmogiais Abstract Our startig poit is a improved versio of a result of. Hajela related to a questio of Komlós: we show that if f) is a fuctio such that lim f) = ad f) = o), there exists 0 = 0f) such that for every 0 ad ay S { 1, 1} with cardiality S 2 /f) oe ca fid orthoormal vectors x 1,..., x R that satisfy 1x x c log f) for all 1,..., ) S. We obtai aalogous results i the case where x 1,..., x are idepedet radom poits uiformly distributed i the Euclidea uit ball B2 or ay symmetric covex body, ad the l -orm is replaced by a arbitrary orm o R. 1 Itroductio Give a pair K, of symmetric covex bodies i R, the parameter βk, ) is the smallest r > 0 such that for ay x 1,..., x K there exist sigs 1,..., { 1, 1} for which 1 x x r. A geeral lower boud for βk, ) was proved by Baaszczyk; i [2] he showed that if K ad are two symmetric covex bodies i R the 1.1) βk, ) c K / ) 1/ for a absolute costat c > 0, where K deotes the volume of K. I what follows we write B p for the uit ball of l p = R, p ), 1 p. A well-kow theorem of Specer [18] states that βb, B ) c, where c > 0 is a absolute costat the same result was proved idepedetly by Gluski i [9]): there exists a absolute costat c > 0 such that for ay 1 ad x 1,..., x R with x i 1, we may fid 1,..., { 1, 1} such that 1.2) 1 x x c. From 1.1) oe readily sees that this result is optimal up to absolute costats. A well-kow questio of Komlós see [19] ad [20]) asks if the sequece βb 2, B ) is bouded. Sice B B 2, a positive aswer to this questio immediately implies Specer s boud. The best kow estimate is due to Baaszczyk [3]: there exists a absolute costat c > 0 such that for ay 1 ad x 1,..., x R with x i 2 1 oe may fid 1,..., { 1, 1} such that 1.3) 1 x x c log. I fact, Baaszczyk proved a more geeral theorem: if K is a covex body i R with Gaussia measure γ K) 1/2 the βb 2, K) C, where C > 0 is a absolute costat; this implies 1.3) because γ rb ) 1/2 for all r c log. While the method of [3] is o-costructive, a algorithmic proof of the O log ) boud i the above statemet was recetly give by Basal, adush ad Garg [6]. The startig poit of this ote is a result of Hajela [11] i the directio of providig a egative aswer to the questio of Komlós. 1

2 Theorem 1.1 Hajela). Let f) be a fuctio such that lim f) = ad f) = o). For ay 0 < λ < 1 2 there exists 0 = 0 f, λ) such that for every 0 ad ay S { 1, 1} with cardiality S 2 /f) oe ca fid orthoormal vectors x 1,..., x R that satisfy ) λ log log f) 1 x x exp log log log f) for all 1,..., ) S. I fact, Hajela cojectures i [11] that the questio of Komlós has a egative aswer ad that the estimate 1.3) that was later obtaied by Baaszczyk should be optimal. Our first result is a improved versio of Theorem 1.1. Theorem 1.2. There exists a absolute costat c 0, 1) such that the followig holds true: For every 1 ad 1 < δ < 1, ad for ay S { 1, 1} with S 2 δ, there exist orthoormal vectors x 1,..., x i R such that i x i c log1/δ) for all 1,..., ) S. Theorem 1.2 implies a stroger versio of Hajela s theorem. Let f) be a fuctio such that lim f) = ad f) = o). Note that if we let δ = δf, ) = f) 1 i Theorem 1.2 the 1 < δ < 1 for large eough, resultig to the lower boud i x i c log f), which improves upo the estimate of Theorem 1.1. I the proof of Theorem 1.2, preseted i Sectio 2 below, iitially we follow the idea of Hajela: the vectors x 1,..., x are obtaied from a radom rotatio of the stadard basis e 1,..., e of R. The improvemet o the estimates is due to Lemma 2.1 which provides a stroger small-ball probability estimate for the -orm o the sphere. The above method to lower-boud the l -orm of a siged sum of vectors has obvious limitatios. Namely, oe should cosider a subset S { 1, 1} of cardiality 2 o) for Theorem 1.2 to obtai a lower boud of order greater tha the oe i 1.1). Thus, this lie of thikig by itself does ot seem adequate to provide a egative aswer to the questio of Komlós. Nevertheless, we fid the lik betwee small ball probability estimates ad the orm of siged sums of vectors iterestig i its ow right, so i Sectio 3 we further explore this pheomeo i several aspects. Iitially, the vectors x 1,... x are assumed to satisfy a lower boud of the form ix i 2 c for all choices of sigs i = ±1, ad the l -orm is replaced by a arbitrary orm o R. I particular, give a symmetric covex body i R, we will prove that for x 1,... x as above ad for ay big subset S { 1, 1}, the -orm of iux i is large for every 1,..., ) S, with overwhelmig probability with respect to U O). This largeess is determied by the Gaussia measure of dilates of ; to make this more precise, we give the followig defiitio. efiitio 1.3. Give a symmetric covex body i R, ad δ 0, 1), let t,δ := max{t > 0 : γ 2t m)) 2 δ e) }, where m) is the media of with respect to the stadard Gaussia measure γ o R. I the case that is the uit ball B p of some l p, p [1, ], we abbreviate t p,δ := t B p,δ. Note that, for ay symmetric covex body i R ad δ 0, 1), t,δ satisfies the bouds 1.4) c 1 1/ t,δ m) c 2 m), 2

3 for some absolute costats c 1, c 2 > 0 for completeess, we provide a short justificatio of 1.4) i Sectio 3.1). Although the defiitio of t,δ might seem somewhat artificial at first sight, we trust that the idea behid it will become clear i the sequel see the commets after Theorem 1.5). Usig the otatio itroduced above, our first result is the followig statemet. Theorem 1.4. Let be a symmetric covex body i R ad δ 0, 1). For ay τ > 0, ay -tuple of vectors x 1,..., x with mi ix i 2 τ ad ay S { 1, 1} with S 2 δ, there is some i=±1 U O) with ν U) 1 e such that for every U U, i Ux i ) τt,δ m) holds for all = 1,..., ) S. We the cosider the case where x 1,..., x are poits i a arbitrary covex body K i R. We observe that a alterative proof of 1.1) ca be deduced from a more geeral result of Gluski ad V. Milma i [10]: Let be a star body i R with 0 it) ad V 1,..., V m be measurable subsets of R with = V 1 = = V m. For ay λ 1,..., λ m R ad ay 0 < t < 1 oe has P x i ) m m, x i V i : λ i x i t m λ 2 i ) 1/2 ) ) te 1 t2 2, where the probability is with respect to the product of the uiform probability measures µ i A) = A Vi V i. The proof of this estimate i [10] is based o a rearragemet iequality of Brascamp, Lieb ad Luttiger [5]. As a ext step, usig Theorem 1.4 we obtai the followig variat of the result of Gluski ad Milma, i the case that V i = B 2 for all i. Theorem 1.5. Let δ 0, 1), be a symmetric covex body i R ad S { 1, 1} with S 2 δ. The ) P x i ) B2 : i x i 1 10 t,δm), for some S 2e. This theorem may be viewed as a extesio of Hajela s result: the l -orm is replaced by a arbitrary orm o R ad the statemet holds for a radom choice of vectors i the Euclidea uit ball. I this cotext, the role of the parameter t,δ i the statemet of our results becomes ow more trasparet: Sice, by 1.4), t,δ m) t,δ M) t,δ M)vrad) 1/, where M) = S 1 x dσx) ad vrad) = / B 2 ) 1/, ad sice M)vrad) 1 this is a simple cosequece of Hölder s iequality), we see that Theorem 1.5 provides iformatio stroger tha the oe from 1.1) provided that M)vrad) 1 ad/or t,δ 1. This is the case for the -orm ad it would be iterestig to provide further examples with sharp estimates. The fuctio t γ 2t m)) that appears i the defiitio of t,δ above has bee studied i [12], [13] ad [16] ad elsewhere. A sample of estimates is the followig: I [12] it is proved that for every 0 < t < 1 2 oe has γ {x : x t m)}) 1 2 td), where { )} m) d) = mi, log γ. 2 3

4 I [13] it is proved that if γ ) 1 2 the for every 0 < t < 1 2 oe has γ t) 2t) r2 )/4 γ ) where r) is the iradius of. I [16] it is proved that for every 0 < t < 1 2 oe has γ {x : x t m)}) 1 2 tc/β), where ad c > 0 is a absolute costat. β) = Var γ ) M 2 ) I this ote we discuss the estimates that ca be derived from the above i special istaces, as for example whe is a l p ball. Fially, the argumet i the proof of Theorem 1.5 ca be further geeralized to the case where x 1,..., x are idepedet radom poits chose uiformly from ay symmetric covex body. Theorem 1.6. There exists a absolute costat c > 0 such that the followig holds: Let δ 0, 1), be a symmetric covex body i R ad S { 1, 1} with S 2 δ. For ay symmetric covex body K i R with K = B2 we may fid U O) with ν U) > 1 2e /2 such that, for ay U U, ) P z i ) UK) UK) : i z i ct,δ m), for some S e /2. 2 A improved versio of Hajela s result I this sectio we preset the proof of Theorem 1.2. I what follows, e 1,..., e is the stadard basis of R. We deote by S 1 the Euclidea sphere i R, ad by σ the uique rotatioally ivariat probability measure o S 1. Recall that σ ca be defied via the Haar probability measure ν o the orthogoal group O) as follows: For every measurable A S 1 we have σa) = ν {U O) : Ue 1 ) A}). We eed a small ball probability estimate for the l -orm, which is give i the ext lemma. It provides a boud whose behaviour for small values of t will play a crucial role i the sequel; i the case of this type of behaviour has bee observed i various works see e.g. [21] ad [15]). We provide a direct ad short proof of the exact statemet that we eed. Lemma 2.1. There exists a absolute costat c 0, 1) such that, for ay 1 ad δ 1, 4) 1, { }) σ θ S 1 : θ c log1/δ) < 2 δ, where σ is the rotatioally ivariat probability measure o S 1. Proof. We use the fact see [12] for its simple proof) that if A is a symmetric covex body i R the σs 1 A) 2γ 2 A), to write { σ θ S 1 : θ s }) 2 2γ {x R : x s}) = Φs))) 2 exp 21 Φs))), 4

5 where, as usual, Φ stads for the cumulative distributio fuctio of the stadard ormal distributio, that is, Φx) = 2π) 1/2 x e t2 /2 dt. Usig the iequality t + s) 2 2t 2 + s 2 ), we have that, for ay s > 0, It follows that Fially, if δ 1, Φs) = 1 2π which proves the desired statemet. { σ θ S 1 : θ 0 2π e t+s)2 2 dt e s2 ) the choosig s = 1 2 log ) 1 δ we get 0 e t2 dt = e s s }) 2 2 exp ) 2 e s2. 2 exp 2 ) e s2 = 2 exp 2 ) δ 1/4 < 2 δ, Proof of Theorem 1.2. The idea of the proof is the same as i Hajela s paper. We cosider orthoormal systems obtaied as radom rotatios of the stadard basis e 1,..., e of R. For ay = 1,..., ) { 1, 1}, there is some V O) such that e i = V ie i ). Thus, by the rotatioal ivariace of the Haar measure, we have, for ay α > 0, ) }) { ν {U O) : U ) }) i e i α = ν U O) : U e i α. Let δ 1, 1/4). Sice 1/2 e i S 1, by the defiitio of the measure σ o S 1 it follows, takig α = c log1/δ) where c is the costat i Lemma 2.1, that ) ν {U O) : U i e i c }) log1/δ) = ν {U O) : U e i { = σ }) θ S 1 : θ c log1/δ). ) }) c log1/δ) Applyig Lemma 2.1 we get σ { θ S 1 : θ c log1/δ) )) < 2 δ. This shows that ) 2.1) ν {U O) : U i e i c }) log1/δ) < 2 δ. 5

6 Now let S { 1, 1} with S 2 δ. The, by the uio boud ad 2.1), ν {U O) : i Ue i ) c }) log1/δ), for every S = = 1 ν {U O) : i Ue i ) c }) log1/δ), for some S 1 ν {U O) : i Ue i ) c }) log1/δ) S > 1 S 2 δ 0. We thus deduce that there exists some U 0 O) such that, if we set x i = U 0 e i ), i = 1,...,, the i x i c log1/δ), for ay = 1,..., ) S, give that δ < 1/4. Fially, ote that i the case δ 1/4, 1) the wated result holds trivially, sice for ay U O) ad every { 1, 1}, i Ue i 1 c log 1/δ is valid, for a suitable absolute costat c > 0. 3 Siged sums of radom vectors from a covex body The argumet used for the proof of Theorem 1.2 motivates us to cosider a more geeral settig. As a first step, we let the l orm be replaced by a arbitrary orm i R ad relax our assumptios o the choice of the vectors x 1,..., x B 2. As a secod step, we will cosider a further geeralizatio, lettig x 1,..., x be chose uiformly ad idepedetly from B 2, or ay covex body K. 3.1 Norms of siged sums of vectors Let be a symmetric covex body i R. We cosider the media m) of Z where Z N0, I ) is a stadard Gaussia radom vector i R. Repeatig the argumet of the previous sectio we ca show the followig. Propositio 3.1. Let be a symmetric covex body i R ad τ > 0. Assume that the vectors x 1,..., x R satisfy ζ i x i τ 2, for all ζ i = ±1. The, for ay S { 1, 1} with S γ 2t τ m) ) < 1, we may fid U O) with ν U) > 1 S γ 2t τ m)) such that ay U U satisfies i Ux i ) t m) for all = 1,..., ) S. 6

7 Proof. Let x 1,..., x R be as i the statemet above, ad fix 1,...,. Normalizig by ix i 2 ad usig the fact that the latter is greater tha τ, we get }) ν {U O) : i Ux i ) t m) Therefore, ν {U O) : U ) ix i ix i 2 { = σ θ S 1 : θ t m) }) τ 2t ) γ τ m) = P Z 2t ) τ m). }) t m) τ }) ν {U O) : i Ux i ) t m), for every S 1 }) ν {U O) : i Ux i ) t m) S ) 2t > 1 S γ τ m). This proves the claim of the propositio. Theorem 1.4 ow is a direct corollary of Propositio 3.1. Proof of Theorem 1.4. Let t = τt,δ ad S { 1, 1} with S 2 δ. By the defiitio of t,δ we have that S γ 2t τ m)) < e. We ca thus apply Propositio 3.1 to get U O) such that ν U) 1 e ad i Ux i t m) = τt,δ m) for every U U ad all 1,..., ) S, which is the statemet of the theorem. Before we proceed, let us briefly explai at this poit the geeral bouds o the parameter t,δ, stated i the itroductio. Proof of 1.4). For the upper boud, it is straightforward by the defiitio of the media that γ m)) = γ Z m)) 1 2 > 2δ e), hece t,δ 1 2. For the lower boud, sice, itegratig i polar coordiates, x S 1 Markov s iequality implies that σ x e η B 2 ) 1/ ) e η dσx) = B 2, holds for every η > 0. We ca relate the latter to the Gaussia measure; usig the same argumet as i the proof of [12, Lemma 2.1], oe ca prove that for ay a 1, ) 1 γ e η B 1/ ) ) 2 B ) σ x e η 1/ ) ) γ B 1 a a 2 e η + γ B a 2. 7

8 We ca boud the secod term usig, e.g. [4, Propositio 2.2], to get γ 1 ) a B 2 a exp a 2 1) 2a ). 2 Note that, for η = log4e) we have 2e η = e) 2 δ e) for ay δ 0, 1), so it suffices to have a a 2 ) 1) exp 2a 2 4e) = e η, which is satisfied if a is large eough a = 20 will do). Now, sice γ every δ 0, 1), the defiitio of t,δ implies that for some absolute costat c 1 > 0. t,δ 1 160e m) 3.2 Radom poits from covex bodies 1 80e ) B 1/ 2 1 c 1 1/ m), ) ) B 1/ 2 δ e) for We will see ow that, suitably ormalized, the hypothesis o the orm of the sum ζ 1 x ζ x i the statemet of Propositio 3.1 is satisfied with overwhelmig probability whe the x i s are chose uiformly ad idepedetly at radom from the iterior of a geeral symmetric covex body K. The required lower boud, which actually holds for ay orm i R i place of the Euclidea orm, will be deduced as a corollary of the followig result of Gluski ad V. Milma [10]: Propositio 3.2 Gluski-V. Milma). Let be a star body i R with 0 it) ad V 1,..., V m be measurable subsets of R with = V 1 = = V m. For ay λ 1,..., λ m R ad ay 0 < t < 1 oe has ) P x i ) m m, x i V i : λ i x i t m ) 1/2 ) λ 2 i te 1 t2 2. Below, we will make use of the followig special istace of Propositio 3.2, which also provides a alterative proof of Baaszczyk s geeral lower boud 1.1) for βk, ). Corollary 3.3. Let K ad be symmetric covex bodies i R, ad let x 1,..., x be radom poits chose uiformly from K. The iequality i x i 1 ) 1/ K 10 is the valid for ay choice of 1,..., { 1, 1}, with probability greater tha 1 e with respect to x 1,..., x. Proof. Let t > 0 ad assume first that K =. We have ) P x i ) 1 K : i x i t, for some 1,..., { 1, 1} ) 2 P x i ) 1 K : i x i t Now if we apply Propositio 3.2 for m :=, V i := K ad λ i := 1 i for every i, ad t such that 2te 1 t2 )/2 < e 1 say t = 1/10) we get ) P x i ) 1 K : i x i t, for some 1,..., { 1, 1} 2 te 1 t2 2 ) < e. 8 2

9 We deduce that, with probability greater tha 1 e with respect to x 1,..., x ), we have i x i > 10 1 for every choice of 1,..., { 1, 1}. For a geeral pair of symmetric covex bodies K,, set a = / K ) 1/ ad K = ak. The we ca apply the above for the pair K, to deduce that for every choice of 1,..., { 1, 1}, i ax i > 1, 10 or, equivaletly, i x i > 1 10 ) 1/ K holds with probability greater tha 1 e with respect to x 1,..., x ). Remark 3.4. Note that Corollary 3.3 immediately implies Baaszczyk s lower boud o βk, ): βk, ) c K / ) 1/ Poits from the ball First we deal with the situatio where the vectors x 1,..., x are chose idepedetly ad uiformly at radom from B 2. The statemet of Theorem 1.5 follows immediately from the defiitio of t,δ ad the ext cosequece of Propositio 3.1. This ca be viewed as a geeralizatio of Hajela s result, i the spirit of Propositio 3.2. Theorem 3.5. Let be a symmetric covex body i R ad S { 1, 1}. The ) P x i ) B2 : i x i t m), for some S S γ 20t m)) + e. Proof. Let A B2 ) be the set { A := x i ) B2 : i x 2 i 1 }, for every 10 1,..., { 1, 1}. By Corollary 3.3, applied for K = = B2, we have that PA c ) < e. This fact, combied with Propositio 3.1 gives ) P z i ) B2 : i z i t m), for some S ) = P x i ), U) B2 ) O) : i Ux i ) t m), for some S ) P x i ), U) A O) : i Ux i ) t m), for some S + e { })] [ν U O) : i Ux i ) t m), for some S dµx ) dµx 1 ) + e as claimed. A < S γ 20t m)) + e, 9

10 Applicatio: the case of l p. As a applicatio let us cosider the case = Bp, 1 p. We remark that, although the estimate βb2, B2 ) seems to be well-kow a proof of this ca be foud i [7]), we could ot locate i the literature ay upper boud o the parameter βb2, Bp ), for p 2. However, for 1 p < 2, oe ca use the previously metioed estimate for βb2, B2 ): if we kow that for every x 1,..., x B2 there exists i ) { 1, 1} such that 1x x 2, the oe ca write 1 x x p 1 p x x 2 1/p. The above estimate is actually sharp, by the lower boud 1.1) ad the fact that B 2 / B p ) 1/ the order of 1 p 1 2. O the other had, for the case p > 2 oe could use the estimate of Baasczcyk βb2, B ) c log i a similar fashio, to deduce that βb2, Bp ) c 1/p log. Recall the defiitio of the parameter β i [17]: β) = Var γ ) M 2. ) For = B p, β) has bee computed i [17]: oe has βb p ) 2p p 2 if 1 p c log ad βb p ) log ) 2 if C log p, for some absolute costats c, C > 0 for a more detailed aalysis, oe may cosult [14]). Moreover, from results of Guédo ad Lata la see [8, Sectio 2.4]) it is kow that, i geeral, m) E Z for ay symmetric covex body i R, ad i particular while mb p ) E Z p 1/p p, 1 p log, mb p ) E Z p log, log p. Therefore, choosig t = 1 4 i Theorem 3.5 ad usig the estimate γ {x : x p t mb p )}) 1 2 tc/βb p ), we get: Corollary 3.6. For ay p 1 there exists a costat c p > 0 such that for ay 0 p) ad ay S { 1, 1} with S 2 cp we have that a radom -tuple of poits i B2 satisfies, with probability greater tha 1 e, i x i c p p 1/p for all = 1,..., ) S. Remark 3.7. It is useful to compare the last result with 1.1). For every p log, sice B 2 / B p ) 1/ is of the order of 1 p 1 2 we see that for ay S { 1, 1} with S 2 cp a radom -tuple of poits i B2 satisfies, with probability greater tha 1 e, i x i c p p ) B 1/ 2 Bp for all = 1,..., ) S. O the other had, i the case p > log oe ca use the fact that the orms p ad are equivalet to deduce from Theorem 1.2 that for ay 0 < ϱ < 1 ad S { 1, 1} with S 2 1 ϱ, a radom -tuple of poits i B2 satisfies, with probability greater tha 1 e, i x i cϱ) p log ) B 1/ 2 Bp for all = 1,..., ) S. I fact, the results of this sectio poit out a geeral way to derive variats of 1.1) i this spirit. A meaigful lower boud o ix i ca be obtaied with high probability for the radom -tuple x 1,..., x, if oe is willig to drop the requiremet that this holds for ay = 1,..., ) { 1, 1}, but rather restrict themselves to a big subset S { 1, 1}. is of 10

11 3.2.2 Poits from a symmetric covex body Fially, we study the case where x 1,..., x are chose uiformly ad idepedetly from a arbitrary symmetric covex body K i R. We shall prove the followig geeralizatio of Theorem 3.5, which i tur, for t := ct,δ will give the claim i Theorem 1.6. Theorem 3.8. There exists a absolute costat c > 0 such the followig holds: Let be a symmetric covex body i R ad S { 1, 1}. For ay symmetric covex body K i R let t > 0 be such that S γ ct B 2 / K ) 1/ m)) < e. The, we may fid U O) with ν U) > 1 2e /2 such that, for all U U, ) P z i ) UK) UK) : i z i t m), for some S e /2. Proof. Let { A = x i ) K : i x i > ) 1/ K, for every B2 1,..., { 1, 1}}. By Corollary 3.3, applied for = B2, we have that PA c ) < e. Usig Propositio 3.1 we write ) P z i ) UK) : O) i z i t m), for some S ) = P x i ), U) K O) : i Ux i t m), for some S ) P x i ), U) A O) : i Ux i t m), for some S + e { })] [ν U O) : i Ux i t m), for some S dµx ) dµx 1 ) + e A < S γ 20t B 2 / K ) 1/ m)) + e. Assume that t is chose so that S γ 20t B 2 / K ) 1/ m)) < e. Applyig Markov s iequality we may fid U O) with ν U) > 1 2e /2 such that ) P z i ) UK) : i z i t m), for some S e /2. This proves the theorem. Remark 3.9. Theorem 3.8 illustrates oce more the mai idea behid the improvemet upo Hajela s result, Theorem 1.2, as well as the rest of the results i this ote; small ball probability estimates for the Gaussia measure of covex bodies ca be liked to the deductio of lower bouds for variats of the quatity βk, ), where the -orm of the siged sum of a radom -tuple of vectors ca be lower bouded for every choice of sigs i ay appropriately large subset of { 1, 1}. 11

12 Ackowledgemets. We would like to thak Apostolos Giaopoulos for useful discussios. The research of the first amed author is supported by the Natioal Scholarship Foudatio IKY), sposored by the act Scholarship grats for secod-degree graduate studies, from resources of the operatioal program Mapower evelopmet, Educatio ad Life-log Learig, , co-fuded by the Europea Social Fud ESF) ad the Greek state. The secod amed author is supported by a Ph Scholarship from the Helleic Foudatio for Research ad Iovatio ELIEK); research umber 70/3/ Refereces [1] S. Artstei-Avida, A. Giaopoulos ad V.. Milma, Asymptotic Geometric Aalysis, Part I, Amer. Math. Soc., Mathematical Surveys ad Moographs ). [2] W. Baaszczyk, Balacig vectors ad covex bodies, Studia Math ), [3] W. Baaszczyk, Balacig vectors ad Gaussia measures of -dimesioal covex bodies, Radom Structures Algorithms ), o. 4, [4] A. Barviok, Measure Cocetratio, lecture otes, Uiversity of Michiga 2005), available at math.lsa.umich.edu/~barviok/total710.pdf. [5] H. J. Brascamp, E. H. Lieb ad J. M. Luttiger, A geeral rearragemet iequality for multiple itegrals, J. Fuct. Aal ), [6] N. Basal,. adush, ad S. Garg, A algorithm for Komlós cojecture matchig Baaszczyks boud, I 57th Aual IEEE Symposium o Foudatios of Computer Sciece, FOCS 2016, New Bruswick, NJ, USA, October , [7] I. Báráy, O the power of liear depedecies, Buildig Bridges, Bolyai Society Mathematical Studies, Vol. 19, Spriger, Berli 2008), pp [8] S. Brazitikos, A. Giaopoulos, P. Valettas ad B-H. Vritsiou, Geometry of isotropic covex bodies, Amer. Math. Society, Mathematical Surveys ad Moographs ). [9] E.. Gluski, Extremal properties of orthogoal parallelepipeds ad their applicatios to the geometry of Baach spaces, Math. USSR Sborik ), [10] E. Gluski ad V. Milma, Geometric probability ad radom cotype 2, Geometric Aspects of Fuctioal Aalysis, volume 1850 of Lecture Notes i Math., pages Spriger, Berli, [11]. Hajela, O a Cojecture of Komlós about Siged Sums of Vectors iside the Sphere, Eur. J. Comb. 91): 33 37, 1988). [12] B. Klartag ad R. Vershyi, Small ball probability ad voretzky Theorem, Israel J. Math ), o. 1, [13] R. Lata la ad K. Oleszkiewicz, Small ball probability estimates i terms of widths, Studia Math ), [14] A. Lytova ad K. E. Tikhomirov, The variace of the l p -orm of the gaussia vector, ad voretzkys theorem, preprit 2017). [15] V.. Milma ad G. Schechtma, A isomorphic versio of voretzky s theorem, C.R. Acad. Sci. Paris ), [16] G. Paouris ad P. Valettas, A Gaussia small deviatio iequality for covex fuctios, Aals of Probability ), [17] G. Paouris, P. Valettas ad J. Zi, Radom versio of voretzky s theorem i l p, Stochastic Process. Appl ), o. 10, [18] J. Specer, Six stadard deviatios suffice, Tras. Amer. Math. Soc ), [19] J. Specer, Balacig vectors i the max orm, Combiatorica 61) 1986), [20] J. Specer, Te Lectures o the Probabilistic Method, Society for Idustrial ad Applied Mathematics, Philadelphia, Pe [21] A. Szakowski, O voretzky s theorem o almost spherical sectios of covex bodies, Israel J. Math ),

13 Keywords: covex bodies, balacig vectors, siged sums, Komlós cojecture, radom rotatios, gaussia measure, small ball probability MSC: Primary 52A23; Secodary 46B06, 52A40, Giorgos Chasapis: epartmet of Mathematics, Uiversity of Athes, Paepistimioupolis , Athes, Greece. Nikos Skarmogiais: epartmet of Mathematics, Uiversity of Athes, Paepistimioupolis , Athes, Greece. 13