Geometry of random sections of isotropic convex bodies

Transcription

1 Geometry of radom sectios of isotropic covex bodies Apostolos Giaopoulos, Labrii Hioi ad Atois Tsolomitis Abstract Let K be a isotropic symmetric covex body i R. We show that a subspace F G, k codimesio k = γ, where γ (/, ), satisfies of K F c γ LK(B 2 F ) with probability greater tha exp( ). Usig a differet method we study the same questio for the L q-cetroid bodies Z q(µ) of a isotropic log-cocave probability measure µ o R. For every q ad γ (0, ) we show that a radom subspace F G,( γ) satisfies Z q(µ) F c 2(γ) q B 2 F. We also give bouds o the diameter of radom projectios of Z q(µ) ad usig them we deduce that if K is a isotropic covex body i R the for a radom subspace F of dimesio (log ) 4 oe has that all directios i F are sub-gaussia with costat O(log 2 ). Itroductio A covex body K i R is called isotropic if it has volume K =, its ceter of mass is at the origi (we call these covex bodies cetered ), ad its iertia matrix is a multiple of the idetity matrix: there exists a costat L K > 0 such that (.) x, θ 2 dx = L 2 K K for every θ i the Euclidea uit sphere S. For every cetered covex body K i R there exists a ivertible liear trasformatio T GL() such that T (K) is isotropic. This isotropic image of K is uiquely determied up to orthogoal trasformatios. A well-kow problem i asymptotic covex geometry asks if there exists a absolute costat C > 0 such that (.2) L := max{l K : K is isotropic i R } C for all (see Sectio 2 for backgroud iformatio o isotropic covex bodies ad log-cocave probability measures). Bourgai proved i [5] that L c 4 log, ad Klartag [9] improved this boud to L c 4. A secod proof of Klartag s boud appears i [2]. Recall that the iradius r(k) of a covex body K i R with 0 it(k) is the largest r > 0 for which rb 2 K, while the radius R(K) := max{ x 2 : x K} of K is the smallest R > 0 for which K RB 2. It is ot hard to see that the iradius ad the radius of a isotropic covex body K i R satisfy the bouds c L K r(k) R(K) c 2 L K, where c, c 2 > 0 are absolute costats. I fact, Kaa, Lovász ad Simoovits [7] have proved that (.3) R(K) ( + )L K. Radius of radom sectios of isotropic covex bodies. The first questio that we discuss i this article is to give sharp upper bouds for the radius of a radom ( k)-dimesioal sectio of K. A atural guess is that the followig questio has a affirmative aswer.

2 Questio.. There exists a absolute costat c 0 > 0 with the followig property: for every isotropic covex body K i R ad for every k, a radom subspace F G, k satisfies (.4) R(K F ) c 0 /k LK. It was proved i [23] that if K is a symmetric covex body i R the a radom F G, k satisfies (.5) R(K F ) c(/k) 3/2 M(K), where c > 0 is a absolute costat ad (.6) M(K) := K I the case of a isotropic covex body oe has K = ad (.7) M(K) ( therefore (.5) implies that a radom F G, k satisfies K K x 2 dx. x 2 2dx) /2 = L K, (.8) R(K F ) c (/k) 3/2 L K, where c > 0 is a absolute costat. Our first mai result shows that oe ca have a boud of the order of γ L K whe the codimesio k is greater tha γ. Theorem.2. Let K be a isotropic symmetric covex body i R ad let k. A radom subspace F G, k satisfies (.9) R(K F ) c 0 max{k, LK } with probability greater tha exp( ), where c 0 > 0 is a absolute costat. The proof is give i Sectio 3. Note that Theorem.2 gives o-trivial iformatio whe k >. I this case, writig k = γ for some γ (/, ) we see that (.0) R(K F ) c 0 LK γ with probability greater tha exp( ) o G,( γ). The result of [23] establishes a γ 3/2 -depedece o γ = k/. A stadard approach to Questio. would have bee to combie the low M -estimate with a upper boud for the mea width (.) w(k) := h K (x) dσ(x), S of a isotropic covex body K i R, that is, the L -orm of the support fuctio of K with respect to the Haar measure o the sphere. This last problem was ope for a umber of years. The upper boud w(k) c 3/4 L K appeared i the Ph.D. Thesis of Hartzoulaki [6]. Other approaches leadig to the same boud ca be foud i Pivovarov [32] ad i Giaopoulos, Paouris ad Valettas [5]. Recetly, E. Milma showed i [26] that if K is a isotropic symmetric covex body i R the (.2) w(k) c 3 (log ) 2 L K. 2

3 I fact, it is ot hard to see that his argumet ca be geeralized to give the same estimate i the ot ecessarily symmetric case. The depedece o is optimal up to the logarithmic term. From the sharp versio of V. Milma s low M -estimate (due to Pajor ad Tomczak-Jaegerma [28]; see [2, Chapter 7] for complete refereces) oe has that, for every k, a subspace F G, k satisfies (.3) R(K F ) c 4 /k w(k) with probability greater tha exp( c 5 k), where c 4, c 5 > 0 are absolute costats. Combiig (.3) with E. Milma s theorem we obtai the folowig estimate: Let K be a isotropic symmetric covex body i R. For every k, a subspace F G, k satisfies (.4) R(K F ) c 2(log ) 2 L K k with probability greater tha exp( c 3 k), where c 2, c 3 > 0 are absolute costats. Note that the upper boud of Theorem.2 has some advatages whe compared to (.4): If k is proportioal to (say k γ for some γ (/, )) the Theorem.2 guaratees that R(K F ) c(γ) L K for a radom F G, k. More geerally, for all k we have c6 (log ) 4 (.5) c 0 max{k, } c 2(log ) 2, k ad hece the estimate of Theorem.2 is stroger tha (.4). Nevertheless, we emphasize that our boud is ot optimal ad it would be very iterestig to decide whether (.4) holds true; this would be optimal for all k. Radius of radom sectios of L q -cetroid bodies ad their polars. I Sectio 4 we study the diameter of radom sectios of the L q -cetroid bodies Z q (µ) of a isotropic log-cocave probability measure µ o R. Recall that a measure µ o R is called log-cocave if µ(λa + ( λ)b) µ(a) λ µ(b) λ for ay compact subsets A ad B of R ad ay λ (0, ). A fuctio f : R [0, ) is called log-cocave if its support {f > 0} is a covex set ad the restrictio of log f o it is cocave. It is kow that if a probability measure µ is log-cocave ad µ(h) < for every hyperplae H, the µ is absolutely cotiuous with respect to the Lebesgue measure ad its desity f µ is log-cocave; see [4]. Note that if K is a covex body i R the the Bru-Mikowski iequality implies that the idicator fuctio K of K is the desity of a log-cocave measure. We say that a log-cocave probability measure µ o R is isotropic if its baryceter bar(µ) is at the origi ad x, θ 2 dµ(x) = R for all θ S. Note that the ormalizatio is differet from the oe i (.); i particular, a cetered covex body K of volume i R is isotropic if ad oly if the log-cocave probability measure µ K with desity x L K K/L K (x) is isotropic. The L q -cetroid bodies Z q (µ), q, are defied through their support fuctio ( ) /q (.6) h Zq(µ)(y) :=, y Lq(µ) = x, y q dµ(x), R ad have played a key role i the study of the distributio of liear fuctioals with respect to the measure µ. For every q we obtai sharp upper bouds for the radius of radom sectios of Z q (µ) of dimesio proportioal to, thus extedig a similar result of Brazitikos ad Stavrakakis which was established oly for q [, ]. 3

4 Theorem.3. Let µ be a isotropic log-cocave probability measure o R ad let q. The: (i) If k = γ for some γ (0, ), the, with probability greater tha e c4k, a radom F G, k satisfies (.7) R(Z q (µ) F ) c 5 (γ) q, where c 4 is a absolute costat ad c 5 (γ) = O(γ 2 log 5/2 (c/γ)) is a positive costat depedig oly o γ. (ii) With probability greater tha e, a radom U O() satisfies (.8) Z q (µ) U(Z q (µ)) (c 6 q) B 2, where c 6 > 0 is a absolute costat. The method of proof is based o estimates (from [26] ad []) for the Gelfad umbers of symmetric covex bodies i terms of their volumetric parameters; combiig these geeral estimates with fudametal (kow) properties of the family of the cetroid bodies Z q (µ) of a isotropic log-cocave probability measure µ we provide estimates for the miimal radius of a k-codimesioal sectio of Z q (µ). The, we pass to bouds for the radius of radom k-codimesioal sectios of Z q (µ) usig kow results from [2], [34] ad [24]. We coclude Sectio 4 with a discussio of the same questios for the polar bodies Z q (µ) of the cetroid bodies Z q (µ). Usig the same approach we study the diameter of radom sectios of covex bodies which have maximal isotropic costat. Set (.9) L := max{l K : K is a isotropic symmetric covex body i R }. It is kow that L cl for some absolute costat c > 0 (see [9, Chapter 3]). We prove the followig: Theorem.4. Assume that K is a isotropic symmetric covex body i R with L K = L. The: (i) A radom F G,/2 satisfies (.20) R(K F ) c 7 ad (.2) L K F c 8 with probability greater tha e c9, where c i > 0 are absolute costats. (ii) A radom U O() satisfies (.22) K U(K) (c 0 ) B 2, with probability greater tha e, where c 0 > 0 is a absolute costat. The same argumets work if we assume that K has almost maximal isotropic costat, i.e. L K βl for some (absolute) costat β (0, ). We ca obtai similar results, with the costats c i ow depedig oly o β. It should be oted that Aloso-Gutiérrez, Bastero, Berués ad Paouris [] have proved that every covex body K has a sectio K F of dimesio k with isotropic costat ( e (.23) L K F c k log k For the proof of this result they cosidered a α-regular M-positio of K. I Theorem.4 we cosider covex bodies i the isotropic positio ad the estimates (.20) ad (.2) hold for a radom subspace F. ). 4

5 Radius of radom projectios of L q -cetroid bodies ad sub-gaussia subspaces of isotropic covex bodies. Let K be a cetered covex body of volume i R. We say that a directio θ S is a ψ α -directio (where α 2) for K with costat b > 0 if (.24), θ Lψα (K) b, θ 2, where (.25), θ Lψα (K) := if { t > 0 : exp ( ( x, θ /t) α) } dx 2. K Markov s iequality implies that if K satisfies a ψ α -estimate with costat b i the directio of θ the for all t we have {x K : x, θ t, θ 2 } 2e ta /b α. Coversely, oe ca check that tail estimates of this form imply that θ is a ψ α -directio for K. It is well-kow that every θ S is a ψ -directio for K with a absolute costat C. A ope questio is if there exists a absolute costat C > 0 such that every K has at least oe sub-gaussia directio (ψ 2 -directio) with costat C. It was first proved by Klartag i [20] that for every cetered covex body K of volume i R there exists θ S such that (.26) {x K : x, θ ct, θ 2 } e t 2 [log(t+)] 2a for all t, where a = 3 (equivaletly,, θ Lψ2 (K) C(log ) a, θ 2 ). This estimate was later improved by Giaopoulos, Paouris ad Valettas i [4] ad [5] (see also [3]) who showed that the body Ψ 2 (K) with support fuctio y, y Lψ2 (K) has volume (.27) c ( ) / Ψ2 (K) c 2 log. Z 2 (K) From (.27) it follows that there exists at least oe sub-gaussia directio for K with costat b C log. Brazitikos ad Hioi i [7] proved that if K is isotropic the logarithmic bouds for, θ Lψ2 (K) hold true with probability polyomially close to : For ay a > oe has {, θ Lψ2 (K) C(log ) 3/2 } max log, a L K for all θ i a subset Θ a of S with σ(θ a ) a, where C > 0 is a absolute costat. Here, we cosider the questio if oe ca have a estimate of this type for all directios θ of a subspace F G,k of dimesio k icreasig to ifiity with. We say that F G,k is a sub-gaussia subspace for K with costat b > 0 if (.28), θ Lψα (K) b, θ 2 for all θ S F := S F. I Sectio 5 we show that if K is isotropic the a radom subspace of dimesio (log ) 4 is sub-gaussia with costat b (log ) 2. More precisely, we prove the followig. Theorem.5. Let K be a isotropic covex body i R. If k (log ) 4 the there exists a subset Γ of G,k (log )3 with ν,k (Γ) such that (.29), θ Lψ2 (K) C(log ) 2 L K for all F Γ ad all θ S F, where C > 0 is a absolute costat. A essetial igrediet of the proof is the good estimates o the radius of radom projectios of the L q -cetroid bodies Z q (K) of K, which follow from E. Milma s sharp bouds o their mea width w(z q (K)) (see Theorem 5.). 5

6 2 Notatio ad prelimiaries We work i R, which is equipped with a Euclidea structure,. We deote the correspodig Euclidea orm by 2, ad write B2 for the Euclidea uit ball, ad S for the uit sphere. Volume is deoted by. We write ω for the volume of B2 ad σ for the rotatioally ivariat probability measure o S. We also deote the Haar measure o O() by ν. The Grassma maifold G,k of k-dimesioal subspaces of R is equipped with the Haar probability measure ν,k. Let k ad F G,k. We will deote the orthogoal projectio from R oto F by P F. We also defie B F = B2 F ad S F = S F. The letters c, c, c, c 2 etc. deote absolute positive costats whose value may chage from lie to lie. Wheever we write a b, we mea that there exist absolute costats c, c 2 > 0 such that c a b c 2 a. Also if A, D R we will write A D if there exist absolute costats c, c 2 > 0 such that c A D c 2 A. Covex bodies. A covex body i R is a compact covex subset A of R with oempty iterior. We say that A is symmetric if A = A. We say that A is cetered if the ceter of mass of A is at the origi, i.e. A x, θ dx = 0 for every θ S. The volume radius of A is the quatity vrad(a) = ( A / B 2 ) /. Itegratio i polar coordiates shows that if the origi is a iterior poit of A the the volume radius of A ca be expressed as ( ) / (2.) vrad(a) = θ A dσ(θ), S where θ A = mi{t > 0 : θ ta}. The radial fuctio of A is defied by ρ A (θ) = max{t > 0 : tθ A}, θ S. The support fuctio of A is defied by h A (y) := max { x, y : x A }, ad the mea width of A is the average (2.2) w(a) := h A (θ) dσ(θ) S of h A o S. The radius R(A) of A is the smallest R > 0 such that A RB 2. For otatioal coveiece we write A for the homothetic image of volume of a covex body A R, i.e. A := A / A. The polar body A of a covex body A i R with 0 it(a) is defied by (2.3) A := { y R : x, y for all x A }. The Blaschke-Sataló iequality states that if A is cetered the A A B 2 2, with equality if ad oly if A is a ellipsoid. The reverse Sataló iequality of J. Bourgai ad V. Milma [6] states that there exists a absolute costat c > 0 such that (2.4) ( A A ) / c/ wheever 0 it(a). For every cetered covex body A of volume i R ad for every q (, ) \ {0} we defie ( ) /q (2.5) I q (A) = x q 2 dx. A As a cosequece of Borell s lemma (see [9, Chapter ]) oe has (2.6) I q (A) c qi 2 (A) for all q 2. For basic facts from the Bru-Mikowski theory ad the asymptotic theory of covex bodies we refer to the books [33] ad [2] respectively. 6

7 Log-cocave probability measures. Let µ be a log-cocave probability measure o R. The desity of µ is deoted by f µ. We say that µ is cetered ad we write bar(µ) = 0 if, for all θ S, (2.7) x, θ dµ(x) = R x, θ f µ (x)dx = 0. R The isotropic costat of µ is defied by (2.8) L µ := ( ) supx R f µ (x) [det Cov(µ)] 2, f R µ (x)dx where Cov(µ) is the covariace matrix of µ with etries x R (2.9) Cov(µ) ij := i x j f µ (x) dx x R i f µ (x) dx f R µ (x) dx f R µ (x) dx R x j f µ (x) dx R f µ (x) dx. We say that a log-cocave probability measure µ o R is isotropic if bar(µ) = 0 ad Cov(µ) is the idetity matrix. Note that a cetered covex body K of volume i R is isotropic, i.e. it satisfies (.), if ad oly if the log-cocave probability measure µ K with desity x L K K/L K (x) is isotropic. Note that for every log-cocave measure µ o R oe has (2.0) L µ κl, where κ > 0 is a absolute costat (a proof ca be foud i [9, Propositio 2.5.2]). We will use the followig sharp result o the growth of I q (K), where K is a isotropic covex body i R, proved by Paouris i [29] ad [30]. Theorem 2. (Paouris). There exists a absolute costat δ > 0 with the followig property: if K is a isotropic covex body i R, the (2.) for every q. LK = δ δ I 2(K) I q (K) I q (K) δi 2 (K) = δ L K For every q ad every y R we set ( ) /q (2.2) h Zq(µ)(y) = x, y q dµ(x). R The L q -cetroid body Z q (µ) of µ is the symmetric covex body with support fuctio h Zq(µ). Note that µ is isotropic if ad oly if it is cetered ad Z 2 (µ) = B 2. If K is a isotropic covex body i R we defie Z q (K) = L K Z q (µ K ). From Hölder s iequality it follows that Z (K) Z p (K) Z q (K) Z (K) for all p q, where Z (K) = cov{k, K}. Usig Borell s lemma, oe ca check that (2.3) Z q (K) c q p Z p(k) for all p < q. I particular, if K is isotropic, the R(Z q (K)) c ql K. Oe ca also check that if K is cetered, the Z q (K) c 2 Z (K) for all q. It was show by Paouris [29] that if q the (2.4) w ( Z q (µ) ) q, ad that for all q, (2.5) vrad(z q (µ)) c q. 7

8 Coversely, it was show by B. Klartag ad E. Milma i [2] that if q the (2.6) vrad(z q (µ)) c 2 q. This determies the volume radius of Z q (µ) for all q. For larger values of q oe ca still use the lower boud: (2.7) vrad(z q (µ)) c 2 q L µ, obtaied by Lutwak, Yag ad Zhag i [25] for covex bodies ad exteded by Paouris ad Pivovarov i [3] to the class of log-cocave probability measures. Let µ be a probability measure o R with desity f µ with respect to the Lebesgue measure. For every k ad every E G,k, the margial of µ with respect to E is the probability measure π E (µ) o E, with desity (2.8) f πe (µ)(x) = f µ (y)dy. x+e It is easily checked that if µ is cetered, isotropic or log-cocave, the π E (µ) is also cetered, isotropic or log-cocave, respectively. A very useful observatio is that: (2.9) P F ( Zq (µ) ) = Z q ( πf (µ) ) for every k ad every F G, k. If µ is a cetered log-cocave probability measure o R the for every p > 0 we defie { (2.20) K p (µ) := K p (f µ ) = x : r p f µ (rx) dr f } µ(0). p From the defiitio it follows that K p (µ) is a star body with radial fuctio (2.2) ρ Kp(µ)(x) = ( f µ (0) 0 0 ) /p pr p f µ (rx) dr for x 0. The bodies K p (µ) were itroduced i [3] by K. Ball who showed that if µ is log-cocave the, for every p > 0, K p (µ) is a covex body. If K is isotropic the for every k ad F G, k, the body K k+ (π F (µ K )) satisfies (2.22) K F /k L K k+ (π F (µ K )) L K. For more iformatio o isotropic covex bodies ad log-cocave measures see [9]. 3 Radom sectios of isotropic covex bodies The proof of Theorem.2 is based o Lemma 3. ad Lemma 3.2 below. They exploit some ideas of Klartag from [8]. Lemma 3.. Let K be a isotropic covex body i R. For every k there exists a subset A := A(, k) of G, k with ν, k (A) e that has the followig property: for every F A, (3.) {x K F : x 2 c LK } e (k+ ) K F, where c > 0 is a absolute costat. 8

9 Proof. Itegratio i polar coordiates shows that for all q > 0 (3.2) x k+q 2 dx dν, k (F ) = ( k)ω k ω G, k K F K x q 2 dx = ( k)ω k Iq q (K), ω ad a applicatio of Markov s iequality shows that a radom F G, k satisfies (3.3) x k+q 2 dx ( k)ω k (ei q (K)) q ω K F with probability greater tha e q. Fix a subspace F G, k which satisfies (3.3). From (2.22) we have (3.4) K F /k c 2 L Kk+ (π F (µ K )) L K c 3 L K where c 2, c 3 > 0 are absolute costats. A simple computatio shows that (3.5) ( k)ω k ω (c 4 ) k for a absolute costat c 4 > 0. Usig also (2.) with q = we get x k+ ( k)ω k 2 dx (ei K F K F ω (K)) (3.6) K F where c 6 > 0 is a absolute costat. It follows that (c 5 L K ) k (c 4 ) k (eδ L K ) (c 6 LK ) k+, (3.7) {x K F : x 2 ec 6 LK } e (k+ ) K F. ad the lemma is proved with c = ec 6. The ext lemma comes from [8]. Lemma 3.2 (Klartag). Let A be a symmetric covex body i R m. The, for ay 0 < ε < we have (3.8) {x A : x 2 εr(a)} 2 ( ε)m A. Proof. Let x 0 A such that x 0 2 = R(A) ad defie v = x 0 / x 0 2. We cosider the set A + defied as (3.9) A + := {x A : x, v 0}. Sice A is symmetric, we have A + = A /2. Note that (3.0) {x A : x 2 εr(a)} εx 0 + ( ε)a +. Therefore, (3.) {x A : x 2 εr(a)} εx 0 + ( ε)a + = ( ε) m A + = 2 ( ε)m A, as claimed. Proof of Theorem.2. Let K be a isotropic symmetric covex body i R. Applyig Lemma 3. we fid a subset A of G, k with ν, k (A) e such that, for every F A, (3.2) {x K F : x 2 c LK } e (k+ ) K F. 9

10 We distiguish two cases: Case. If k > /3 the choosig ε 0 = e 3 we get (3.3) 2 ( ε 0) k K F = k 2 e 3 K F > e k 3 K F > e (k+ ) K F, because k + > k 3 +. By Lemma 3.2 ad (3.2) we get that (3.4) {x K F : x 2 ε 0 R(K F )} > {x K F : x 2 c LK }, therefore (3.5) R(K F ) < c 2 LK, where c 2 = ε 0 c > 0 is a absolute costat. Case 2. If k /3 the we choose ε = k+ 6( k). Note that ε < /2. Usig the iequality t > e 2t o (0, /2) we get (3.6) 2 ( ε ) k K F = ( k + ) k K F > e k+ 3 K F > e (k+ ) K F, 2 6( k) because 2(k+ ) 3 >. By Lemma 3.2 this implies that (3.7) {x K F : x 2 ε R(K F )} > {x K F : x 2 c LK }, therefore (3.8) ε R(K F ) < c LK, which, by the choice of ε becomes (3.9) R(K F ) < c 3 max{k, LK } for some absolute costat c 3 > 0. This completes the proof of the theorem (with a probability estimate e for all k ). Remark 3.3. It is possible to improve the probability estimate e i the rage k γ, for ay γ (/, ). This ca be doe with the help of kow results that demostrate the fact that the existece of oe s-dimesioal sectio with radius r implies that radom m-dimesioal sectios, where m < s, have radius of the same order. This was first observed i [2], [34] ad, soo after, i [24]. Let us recall this last statemet. Let A be a symmetric covex body i R ad let s < m. If R(A F ) r for some F G,m the a radom subspace E G,s satisfies ( c2 ) s 2(m s) (3.20) R(A E) r m with probability greater tha 2e ( s)/2, where c 2 > 0 is a absolute costat. We apply this result as follows. Let k = γ ad set t = δ, where δ γ/ log( + /γ). From the proof of Theorem.2 we kow that there exists E G, t such that (3.2) R(K E) c LK, t 0

11 where c > 0 is a absolute costat. Applyig (3.20) with s = k ad m = t we see that a radom subspace F G, k satisfies (3.22) R(K F ) ( c2 ) 3 2 δ R(K E) = c 3 (γ) L K with probability greater tha 2e k/2, where c 3 (γ) = O((γ log( + /γ)) 3 2 ). Remark 3.4. It is also possible to give lower bouds of the order of L K for the diameter of ( k)- dimesioal sectios, provided that the codimesio k is small. Itegratio i polar coordiates shows that (3.23) x q 2 dx = ω x k q 2 dx dν, k (F ) K ( k)ω k G, k K F for every k ad every 0 < q <. It follows that (3.24) x k q 2 dx dν, k (F ) = ( k)ω k I q q (K), ω G, k K F ad a applicatio of Markov s iequality shows that a radom F G, k satisfies (3.25) x k q 2 dx ( k)ω k (e/i q (K)) q ω K F with probability greater tha e q. Assumig that q > k, for ay F G, k satisfyig (3.25) we have (3.26) K F R(K F ) k q x k q 2 dx ( k)ω k (e/i q (K)) q, ω which implies K F ( ) q k ω (3.27) R(K F ) K F q k ( k)ω k ( I q (K) e ) q ( ) k q k c q k (c2 I q (K)) q q k. LK If k the we may choose q = 2 ad use the fact that I 2 (K) c 3 LK by Theorem 2., to get: Propositio 3.5. Let K be a isotropic covex body i R. For every k there exists a subset A of G, k with ν, k (A) e such that, for every F A, (3.28) R(K F ) c L K, where c > 0 is a absolute costat. Remark 3.6. Choosig k = /2 i Theorem.2 we see that if K is a isotropic symmetric covex body i R the a subspace F G, /2 satisfies (3.29) R(K F ) c LK with probability greater tha 2 exp( c 2 ), where c, c 2 > 0 are absolute costats. A stadard argumet that goes back to Krivie (see [2, Propositio 8.6.2]) shows that there exists U O() such that (3.30) K U(K) (c 3 LK ) B 2, where c 3 > 0 is a absolute costat. I fact, oe ca prove a aalogue of (3.30) for a radom U O() usig a result of Vershyi ad Rudelso (see [34, Theorem.]): There exist absolute costats γ 0 (0, /2)

12 ad c > 0 with the followig property: if A ad D are two symmetric covex bodies i R which have sectios of dimesios at least k ad 2γ 0 k respectively whose radius is bouded by, the a radom U O() satisfies (3.3) R(A U(D)) c /k with probability greater tha e. As a applicatio, settig D = A ad k = /2 oe has the followig (see [8]). If (3.32) r A := mi{r(a F ) : dim(f ) = ( γ 0 ) } the R(A U(A)) c 2 r A with probability greater tha e with respect to U O(). Choosig k = γ 0 /2 i Theorem.2 we see that if K is a isotropic symmetric covex body i R the (3.33) r K c 4 LK for some absolute costat c 4 > 0. This gives that a radom U O() satisfies (3.34) K U(K) (c 5 LK ) B 2, with probability greater tha e, where c 5 > 0 is a absolute costat. 4 Miimal ad radom sectios of the cetroid bodies of isotropic log-cocave measures I this sectio we discuss the case of the L q -cetroid bodies Z q (µ) of a isotropic log-cocave probability measure µ o R. Our method will be differet from the oe i the previous sectio. I view of (3.20) we ca give a upper boud for the radius of a radom k-codimesioal sectio of a symmetric covex body A i R if we are able to give a upper boud for the radius of some t-codimesioal sectio of A, where t k. This leads us to the study of the Gelfad umbers c t (A), which are defied by (4.) c t (A) = mi{r(a F ) : F G, t } for every t = 0,...,. It was proved i [] that if A is a symmetric covex body i R the, for ay t =,..., /2 there exists F G, 2t such that (4.2) A F c (e t log + ) w t (A)B2 F, t where (4.3) w t (A) := sup{vrad(a E) : E G,t }. I other words, (4.4) c 2t (A) c t log (e + t ) w t (A). This is a refiemet of a result of V. Milma ad G. Pisier from [27], where a similar estimate was obtaied, with the parameter w t (A) replaced by (the larger oe) (4.5) v t (A) := sup{vrad(p E (A)) : E G,t }. We shall apply this method to the bodies Z q (µ). The mai additioal igrediet is the ext fact, which combies results of Paouris ad Klartag (see [26] or [9, Chapter 5] for precise refereces): 2

13 Theorem 4.. Let µ be a cetered log-cocave probability measure o R. The, for all t ad q we have (4.6) v t (Z q (µ)) = sup{vrad(p E (Z q (µ))) : E G,t } c 0 q t max{ q, t} max E G,t det Cov(π E (µ)) 2t, where c 0 > 0 is a absolute costat. We apply Theorem 4. as follows: for every t /2 ad every E G,t we have that π E (µ) is isotropic, ad hece det Cov(π E (µ)) 2t =. The, (4.7) w t (Z q (µ)) v t (Z q (µ)) c 0 q t max{ q, t}. From (4.4) we get Lemma 4.2. Let µ be a isotropic log-cocave probability measure o R q. The, ad let t /2 ad (4.8) c 2t (Z q (µ)) c 2 (e t log + ) q t t max{ q, t}, where c 2 > 0 is a absolute costat. Let k 4 ad let t < k/2. From Lemma 4.2 we kow that there exists E G, 2t such that (4.9) R(Z q (µ) E) c 2 (e t log + ) q t t max{ q, t}, where c 2 > 0 is a absolute costat. Applyig (3.20) with s = k ad m = 2t we see that a radom subspace F G, k satisfies (4.0) R(Z q (µ) F ) ( c2 ) k 2(k 2t) t R(Z q (µ) E) ( c3 ) t ( k 2t log e + ) q t t t max{ q, k} with probability greater tha 2e k/2, where c 3 > 0 is a absolute costat. I particular, if k = γ we ca choose t = γ/ log(c/γ), for c > e 2, to get the followig. Theorem 4.3. Let µ be a isotropic log-cocave probability measure o R ad let γ (0, ) ad q. If k γ the a radom subspace F G, k satisfies (4.) R(Z q (µ) F ) c(γ) q with probability greater tha 2e γ/2, where c(γ) = O(γ 2 log 5/2 (c/γ)) is a positive costat depedig oly o γ. Next, we apply (3.3): choosig t = γ 0 /2 i (4.8) we see that (4.2) r Zq(µ) = c γ0(z q (µ)) c 4 q for every q, where c 4 = c 4 (γ 0 ) > 0 is a absolute costat. Therefore, we have: Theorem 4.4. Let µ be a isotropic log-cocave probability measure o R ad let q. The, a radom U O() satisfies (4.3) Z q (µ) U(Z q (µ)) (c q) B 2, with probability greater tha e, where c > 0 is a absolute costat. 3

14 Note that Theorem.3 summarizes the cotets of Theorem 4.3 ad Theorem 4.4. Remark 4.5. We ca study the same questio for the polar body Z q (µ) of Z q (µ). Note that (4.4) w t (Z q (µ)) := sup{vrad(z q (µ) E) : E G,t } [if{vrad(p E (Z q (µ))) : E G,t }] by duality ad by the Bourgai-Milma iequality. For ay t ad ay symmetric covex body A i R defie (4.5) v t (A) = if{vrad(p E (A)) : E G,t }. I the case A = Z q (µ) this parameter has bee studied i []: Lemma 4.6. Let µ be a isotropic log-cocave probability measure o R. For ay q ad k we have: (4.6) v k (Z q(µ)) c mi(q, k). If we assume that sup L α the we have (4.7) v k (Z q(µ)) c 2 mi(q, k) α These estimates are leadig to the ext bouds o the miimal radius of a k-codimesioal sectio of Z q (µ). The followig theorem is also from []. Theorem 4.7. Let µ be a isotropic log-cocave probability measure o R. For ay q ad k we have: (i) There exists F G, k such that: (4.8) P F (Z q (µ)) R k,q B 2 F ad hece R(Z q (µ) F ) R k,q, where (4.9) R k,q = mi {, c 3 (e mi(q /2, k /4 ) k log + ) }. k (ii) If we assume that sup L α the there exists F G, k such that: (4.20) P F (Z q (µ)) R k,q,α B 2 F ad hece R(Z q (µ) F ) R k,q,α, where { (4.2) R k,q,α = mi, c 4 α (e mi(q, k) k log + ) }. k Assumig that q ad choosig k = γ 0 we see from (4.8) ad (4.9) that (4.22) c γ0(z q (µ)) c (γ 0 ) q where c (γ 0 ) > 0 is a absolute costat. The, we apply (3.20) with s = /2 ad m = ( γ 0 ) to get that a radom subspace E G,/2 satisfies (4.23) R(Z q (µ) E) c 3 c γ0(z q (µ)) c 2 (γ 0 ) q 4

15 with probability greater tha 2e /4, where c 2 (γ 0 ) > 0 is a absolute costat. As usual, this implies that a radom U O() satisfies (4.24) Z q (µ) U(Z q (µ)) c q B 2, with probability greater tha e, where c > 0 is a absolute costat. This estimate appears i [22] (ad a secod proof is give i [8]). Assumig that sup L α we may apply the same reasoig for every q : choosig k = γ 0 we see from (4.20) ad (4.2) that (4.25) c γ0(z q (µ)) c (γ 0 ) α q, where c (γ 0 ) > 0 is a absolute costat. The, we apply (3.20) with s = /2 ad m = ( γ 0 ) to get that a radom subspace E G,/2 satisfies (4.26) R(Z q (µ) E) c 3 c γ0(z q (µ)) c 2 (γ 0 ) α q with probability greater tha 2e /4, where c 2 (γ 0 ) > 0 is a absolute costat. Fially, this implies that a radom U O() satisfies (4.27) Z q (µ) U(Z q (µ)) cα q B 2, with probability greater tha e, where c > 0 is a absolute costat. 4. Radom sectios of bodies with maximal isotropic costat Startig with a isotropic symmetric covex body K i R we ca use the method of this sectio i order to estimate the quatities (4.28) c t (K) = mi{r(k F ) : F G, t } for every t = 0,...,. From (2.22) we have L Kk+ (π (4.29) K E t E (µ K )) c2 for every E G,t, therefore L K c 3L t L K ( ) t c3 t L t (4.30) w t (K) c 4 t. L K Assume that K has maximal isotropic costat, i.e. L K = L (the same argumet works if we assume that L K is almost maximal, i.e. L K βl for some absolute costat β (0, )). It is kow that L t c L c 2 L for all t, where c, c 2 > 0 are absolute costats. Therefore, we get: Lemma 4.8. Let K be a isotropic symmetric covex body i R such that L K = L, ad let t /2. The, (4.3) c 2t (K) c t ( t log e + ), t t where c > 0 is a absolute costat. 5

16 The, we apply (3.20) with s = /2 ad m = ( γ 0 ) to get that a radom subspace E G,/2 satisfies (4.32) R(K E) c 3 c γ0(k) c (γ 0 ) with probability greater tha 2e /4, where c (γ 0 ) > 0 is a absolute costat. Also, sice c γ0(k) c(γ 0 ), we may apply (3.3) to get: Theorem 4.9. Let K be a isotropic symmetric covex body i R with L K = L. A radom U O() satisfies (4.33) K U(K) (c 3 ) B 2, with probability greater tha e, where c 3 > 0 is a absolute costat. We ca also prove the local aalogue of this fact: radom proportioal sectios of a body with maximal isotropic costat have bouded isotropic costat. Theorem 4.0. Let K be a isotropic symmetric covex body i R with L K = L. A radom F G,/2 satisfies (4.34) L K F c 4 with probability greater tha e c5, where c 4, c 5 > 0 are absolute costats. Proof. It was proved i [0] (see also [9, Lemma 6.3.5]) that if L K = L the (4.35) K F c6 for every G,/2, where c 6 > 0 is a absolute costat. Sice R(K F ) c 3 for a radom F G,/2, for all these F we get (4.36) 2 L2 K F x 2 2dx R 2 (K F ) c 2 K F + 2 K F 2 6 c2 3, which implies that K F (4.37) L K F c 4, where c 4 = 2c 6 c 3. 5 Sub-Gaussia subspaces I this sectio we prove Theorem.5. We will use E. Milma s estimates [26] o the mea width w(z q (K)) of the L q -cetroid bodies Z q (K) of a isotropic covex body K i R. Theorem 5. (E. Milma). Let K be a isotropic covex body i R. The, for all q oe has { q log( + q) (5.) w(z q (K)) c log( + q) max, } q L K where c > 0 is a absolute costat. We also use the ext fact o the diameter of k-dimesioal projectios of symmetric covex bodies (see [2, Propositio 5.7.]). 6

17 Propositio 5.2. Let D be a symmetric covex body i R ad let k < ad α >. The there exists a subset Γ,k G,k with measure ν,k (Γ,k ) e c2α2k such that the orthogoal projectio of D oto ay subspace F Γ,k satisfies (5.2) R(P F (D)) c 3 α max{w(d), R(D) k/}, where c 2 > 0, c 3 > are absolute costats. Combiig Propositio 5.2 with Theorem 5. ad the fact that R(Z q (K)) cql K, we get: Lemma 5.3. Let K be a isotropic covex body i R. Give q defie k 0 (q) by the equatio (5.3) k 0 (q) = log 2 ( + q) max{log 2 ( + q), /q}. The, for every k k 0 (q), a radom F G,k satisfies { q log( + q) (5.4) R(P F (Z q (K))) c α log( + q) max, } q L K with probability greater tha e c2α2 k 0(q), where c, c 2 > 0 are absolute costats. Proof. Sice R(Z q (K)) cql K we see that (5.5) R(Z q (K)) k 0 (q) cq { } log( + q) max log( + q), L K q { q log( + q) = c log( + q) max, } q L K. From Theorem 5. we have a upper boud of the same order for w(z q (K)). The, we apply Propositio 5.2 for Z q (K). Remark 5.4. Note that if s k the the coclusio of Propositio 5.2 cotiues to hold for a radom F G,s with the same probability o G,s ; this is a immediate cosequece of Fubii s theorem ad of the fact that R(P H (D)) R(P F (D)) for every s-dimesioal subspace H of a k-dimesioal subspace F of R. Proof of Theorem.5. We defie q 0 by the equatio (5.6) q 0 log 2 ( + q 0 ) =. Note that q 0 /(log ) 2 ad log( + q 0 ) log. For every 2 q q 0 we have q log 2 ( + q), therefore (5.7) k 0 (q) = log2 ( + q) q c log 2 ( + q 0 ) q 0 for some absolute costat c > 0, because q log 2 ( + q)/q is decreasig for q 4. It follows that (5.8) k 0 (q) c log 4 ( + q 0 ) c 2 (log ) 4 for all 2 q q 0. Now, we fix α > ad defie (5.9) k 0 = c log 4 ( + q 0 ). 7

18 Usig Lemma 5.3 ad Remark 5.4, for every q q 0 we ca fid a set Γ q G,k0 with ν,k0 (Γ q ) e cα2 k 0 such that { q log( + q) (5.0) R(P F (Z q (K))) c 3 α log( + q) max, } q L K c 3 α q log( + q)l K for all F G,k0. If Γ := log 2 q 0 s= Γ 2 s, the ( (5.) ν,k0 G,k0 \ Γ ) log 2 ν,k0 (G,k0 \ s= Γ 2 s ) c(log )e cα2 k 0 log3 if α is chose large eough. The for every F Γ, for all θ S F ad for every s log 2 q 0 we have (5.2) h Z2 s (K)(θ) 2 s = h P F (Z 2 s (K))(θ) 2 s c 3 α log( + 2 s )L K c 4 α(log )L K. Takig ito accout the fact that if 2 s q < 2 s+ the (5.3) h Zq(K)(y) q h Z 2 s+ (K)(y) 2 s/2 = 2 h Z 2s+ (K)(y) 2 (s+)/2, we see that (5.4) h Zq(K)(y) q c 5 α(log )L K for every F Γ, for all θ S F ad for every 2 q q 0. Next, observe that if q 0 q the we may write (5.5) h Zq(K)(y) q c 6q h Zq0 (K) (y) = c 6 q h Zq0 (K)(y) c 6 q 0 q q0 q0 = c 6 log( + q 0 ) h Z q0 (K)(y) q0 q0 c 7 (log ) h Z q0 (K)(y) q0, h Zq0 (K)(y) q0 ad hece (5.6) h Zq(K)(y) q c 7 α(log ) 2 L K for every F Γ, for all θ S F ad for every q 0 q. Recall that Ψ 2 (K) is the covex body with support fuctio h Ψ2(K)(y) =, y Lψ2 (K). Oe also has h Zq(K)(y) h Zq(K)(y) (5.7) h Ψ2(K)(y) sup sup q 2 q 2 q q because h Zq(K)(y) h Z(K)(y) for all q. The, (5.4) ad (5.6) ad the fact that α show that (5.8), θ Lψ2 (K) C(log ) 2 L K for every F Γ ad for all θ S F, where C > 0 is a absolute costat. 8

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20 [25] E. Lutwak, D. Yag ad G. Zhag, L p affie isoperimetric iequalities, J. Differetial Geom. 56 (2000), -32. [26] E. Milma, O the mea width of isotropic covex bodies ad their associated L p-cetroid bodies, It. Math. Res. Not. IMRN (205), o., [27] V. D. Milma ad G. Pisier, Gaussia processes ad mixed volumes, Aals of Probability 5 (987), [28] A. Pajor ad N. Tomczak-Jaegerma, Subspaces of small codimesio of fiite dimesioal Baach spaces, Proc. Amer. Math. Soc. 97 (986), [29] G. Paouris, Cocetratio of mass i covex bodies, Geometric ad Fuctioal Aalysis 6 (2006), [30] G. Paouris, Small ball probability estimates for log-cocave measures, Tras. Amer. Math. Soc. 364 (202), [3] G. Paouris ad P. Pivovarov, A probabilistic take o isoperimetric-type iequalities, Adv. Math. 230 (202), [32] P. Pivovarov, O the volume of caps ad boudig the mea-width of a isotropic covex body, Math. Proc. Cambridge Philos. Soc. 49 (200), [33] R. Scheider, Covex Bodies: The Bru-Mikowski Theory, Secod expaded editio. Ecyclopedia of Mathematics ad Its Applicatios 5, Cambridge Uiversity Press, Cambridge, 204. [34] R. Vershyi, Isoperimetry of waists ad local versus global asymptotic covex geometries (with a appedix by M. Rudelso ad R. Vershyi), Duke Mathematical Joural 3 (2006), -6. Keywords: Covex bodies, isotropic positio, log-cocave measures, cetroid bodies, diameter of radom sectios, sub-gaussia estimates. 200 MSC: Primary 52A23; Secodary 46B06, 52A40, 60D05. Apostolos Giaopoulos: Departmet of Mathematics, Uiversity of Athes, Paepistimioupolis 57-84, Athes, Greece. apgiaop@math.uoa.gr Labrii Hioi: Departmet of Mathematics, Uiversity of Athes, Paepistimioupolis 57-84, Athes, Greece. lamchioi@math.uoa.gr Atois Tsolomitis: Departmet of Mathematics, Uiversity of the Aegea, Karlovassi , Samos, Greece. atsol@aegea.gr 20