Small ball probability estimates,

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1 Small ball probability estimates, ψ -behavior ad the hyperplae cojecture Nikos Dafis ad Grigoris Paouris Abstract We itroduce a method which leads to upper bouds for the isotropic costat. We prove that a positive aswer to the hyperplae cojecture is equivalet to some very strog small probability estimates for the Euclidea orm o isotropic covex bodies. As a cosequece of our method, we obtai a alterative proof of the result of J. Bourgai that every ψ -body has bouded isotropic costat, with a slightly better estimate: If K is a covex body i R such that, θ q β, θ for every θ S ad every q, the L K Cβ log β, where C > 0 is a absolute costat. Itroductio A covex body K i R is called isotropic if it has volume K =, ceter of mass at the origi, ad its iertia matrix is a multiple of the idetity. Equivaletly, if there is a costat L K > 0 such that (.) x, θ dx = L K K for every θ i the Euclidea uit sphere S. It is ot hard to see that for every covex body K i R there exists a affie trasformatio T of R such that T (K) is isotropic. Moreover, this isotropic image is uique up to orthogoal trasformatios; cosequetly, oe may defie the isotropic costat L K as a ivariat of the affie class of K. The isotropic costat is closely related to the hyperplae cojecture (also kow as the slicig problem) which asks if there exists a absolute costat c > 0 such that max θ S K θ c for every covex body K of volume i R with ceter of mass at the origi. This is because, by Bru s priciple, for ay covex body K i R ad ay θ S, the fuctio t K (θ + tθ) is cocave o its support, ad this implies that (.) x, θ dx K θ. K

2 Usig this relatio oe ca check that a affirmative aswer to the slicig problem is equivalet to the followig statemet: There exists a absolute costat C > 0 such that L K C for every covex body K. We refer to the article [6] of Milma ad Pajor for backgroud iformatio about isotropic covex bodies. The isotropic costat ad the hyperplae cojecture ca be studied i the more geeral settig of log-cocave measures. Let f : R R + be a itegrable fuctio with f(x)dx =. We say that f is isotropic if f has ceter of mass at R the origi ad (.3) x, θ f(x)dx = R for every θ S. It is well-kow that the hyperplae cojecture for covex bodies is equivalet to the followig statemet: There exists a absolute costat C > 0 such that, for every isotropic log-cocave fuctio f o R, (.4) f(0) / C. It is kow that L K L B c > 0 for every covex body K i R (we use the letters c, c, C etc. to deote absolute costats). Bourgai proved i [3] that L K c 4 log ad, a few years ago, Klartag [8] obtaied the estimate L K c 4. The approach of Bourgai i [3] is to reduce the problem to the case of covex bodies that satisfy a ψ -estimate (with costat β = O( 4 )). We say that K satisfies a ψ -estimate with costat β if (.5), y ψ β, y for all y R. Bourgai proved i [4] that, if (.5) holds true, the (.6) L K Cβ log β. The purpose of this paper is to itroduce a differet method which leads to upper bouds for L K. We prove that a positive aswer to the hyperplae cojecture is equivalet to some very strog small probability estimates for the Euclidea orm o isotropic covex bodies; for < p, p 0, we defie ( ) /p (.7) I p (K) := x p dx ad, for δ, we cosider the parameter (.8) q c (K, δ) := max{p : I (K) δi p (K)}. The, the hyperplae cojecture is equivalet to the followig statemet: K There exist absolute costats C, ξ > 0 such that, for every isotropic covex body K i R, q c (K, ξ) C.

3 The mai results of [] ad [3] show that there exists a parameter q := q (K) (related to the L q cetroid bodies of K) with the followig properties: (i) q (K) c, (ii) q c (K, ξ) q (K) for some absolute costat ξ, ad hece, I (K) ξi q (K). The questio that arises is to uderstad what happes with I p (K) whe p lies i the iterval [q, ], where there are o geeral estimates available up to ow. I the case where K is a ψ -body, oe has q ad the problem is automatically resolved. The mai idea i our approach is to start from a extremal isotropic covex body K i R with maximal isotropic costat L K L := sup{l K : K is a covex body i R }. Buildig o ideas from the work [5] of Bourgai, Klartag ad Milma, we costruct a secod isotropic covex body K which is also extremal ad, at the same time, is i α-regular M-positio i the sese of Pisier (see [4]). The, we use the fact that small ball probability estimates are closely related to estimates o coverig umbers. This gives the estimate (.9) L K I c ( α)t α (K ) Ct, for t C(α), where c, C > 0 are absolute costats. The costructio of K from K ca be doe iside ay subclass of isotropic log-cocave measures which is stable uder the operatios of takig margials or products. This leads us to the defiitio of a coheret class of probability measures (see Sectio 4): a subclass U of the class of probability measures P is called coheret if it satisfies two coditios:. If µ U is supported o R the, for all k ad F G,k, π F (µ) U.. If m N ad µ i U, i =,..., m, the µ µ m U. It should be oted that the class of isotropic covex bodies is ot coheret. This is the reaso for workig with the more geeral class of log-cocave measures. The basic tools that eable us to pass from oe laguage to the other come from K. Ball s bodies ad are described i Sectio. Our mai result is the followig: Theorem.. Let U be a coheret subclass of isotropic log-cocave measures ad let ad δ. The, ( ) (.0) sup f µ (0) Cδ sup µ U [] µ U [] q c (µ, δ) log e, q c (µ, δ) where C > 0 is a absolute costat ad U [] deotes the subclass of -dimesioal measures i U. Sice oe has that q c (µ, c) for ay log-cocave isotropic measure i R (where c > 0 is a absolute costat), the Theorem. has the followig cosequece: For every isotropic log-cocave measure i R, (.) f µ (0) C 4 log. 3

4 Moreover, i Sectio 4, for every α (, ] we itroduce a coheret class P α (β), of isotropic log-cocave measures which is cotaied i the class of ψ α -measures with costat β. The, from Theorem. we get: Theorem.. Let α (, ], let β > 0 ad µ (P α (β) IL) []. The, ( (.) f µ (0) C α β log α α β ), α where C > 0 is a absolute costat. For the special case that α =, we prove that for symmetric measures the coheret class P (β) is essetially the same with the ψ -class. The by Theorem. we have that: if µ is a symmetric log-cocave ψ measure with costat β > 0, the (.3) f µ (0) Cβ log β. From Theorem. ad. we immediately deduce two facts:. If a symmetric covex body K satisfies a ψ -estimate with costat β, the. For every covex body K i R, L K Cβ log β. L K C 4 log. The first fact slightly improves Bourgai s estimate from [4]. The secod oe is weaker tha Klartag s 4 -boud i [8]; evertheless, our method has the advatage that it ca take ito accout ay additioal iformatio o the ψ α behavior of K. Ackowledgmet. We would like to thak Apostolos Giaopoulos for may iterestig discussios. Also, the secod amed author wats to thak Assaf Naor for several valuable commets o a earlier versio of this paper. Backgroud material. Basic otatio. We work i R, which is equipped with a Euclidea structure,. We deote by the correspodig Euclidea orm, ad write B for the Euclidea uit ball, ad S for the uit sphere. Volume is deoted by. We write ω for the volume of B ad σ for the rotatioally ivariat probability measure o S. The Grassma maifold G,k of k-dimesioal subspaces of R is equipped with the Haar probability measure µ,k. Let k ad F G,k. We will deote by P F the orthogoal projectio from R oto F. 4

5 The letters c, c, c, c etc. deote absolute positive costats which may chage from lie to lie. I order to facilitate readig, we will deote by c, η, κ, ξ, τ etc. some (absolute) positive costats that appear i more tha oe places. Wheever we write a b, we mea that there exist absolute costats c, c > 0 such that c a b c a. Also if K, L R we will write K L if there exist absolute costats c, c > 0 such that c K L c K.. Probability measures. We deote by P [] the class of all probability measures i R which are absolutely cotiuous with respect to the Lebesgue measure. We write A for the Borel σ-algebra i R. The desity of µ P [] is deoted by f µ. The subclass SP [] cosists of all symmetric measures µ P [] ; µ is called symmetric if f µ is a eve fuctio o R. The subclass CP [] cosists of all µ P [] that have ceter of mass at the origi; so, µ CP [] if (.) R x, θ dµ(x) = 0 for all θ S. Let µ P []. For every k ad F G,k, we defie the F -margial π F (µ) of µ as follows: for every A A F, (.) π F (µ)(a) := µ(p F (A)). It is clear that π F (µ) P [dim F ]. Note that, by the defiitio, for every Borel measurable fuctio f : R [0, ) we have (.3) f(x) dπ F (µ)(x) = f(p F (x)) dµ(x). F R The desity of π F (µ) is the fuctio (.4) π F (f µ )(x) := f πf (µ)(x) = f µ (y) dy. x+f Let µ P [] ad µ P []. We will write µ µ for the measure i P [+ ] which satisfies (.5) (µ µ )(A A ) = µ (A )µ (A ) for all A A ad A A. It is easily checked that f µ µ = f µ f µ..3 Log-cocave measures. We deote by L [] the class of all log-cocave probability measures o R. A measure µ o R is called log-cocave if for ay A, B A ad ay λ (0, ), (.6) µ(λa + ( λ)b) µ(a) λ µ(b) λ. 5

6 A fuctio f : R [0, ) is called log-cocave if log f is cocave o its support {f > 0}. It is kow that if µ L [] ad µ(h) < for every hyperplae H, the µ P [] ad its desity f µ is log-cocave (see []). As a applicatio of the Prékopa-Leidler iequality ([0], [5], [6]) oe ca check that if f is log-cocave the, for every k ad F G,k, π F (f) is also log-cocave. As before, we write CL [] or SL [] for the cetered or symmetric o degeerate µ L [] respectively..4 Covex bodies. A covex body i R is a compact covex subset C of R with o-empty iterior. We say that C is symmetric if x C implies that x C. We say that C has ceter of mass at the origi if C x, θ dx = 0 for every θ S. The support fuctio h C : R R of C is defied by h C (x) = max{ x, y : y C}. The mea width of C is defied by (.7) W (C) = h C (θ)σ(dθ). S For each < p <, p 0, we defie the p-mea width of C by ( ) /p (.8) W p (C) = h p C (θ)σ(dθ). S The radius of C is the quatity R(C) = max{ x : x C} ad, if the origi is a iterior poit of C, the polar body C of C is (.9) C := {y R : x, y for all x C}. Note that if K is a covex body i R the the Bru-Mikowski iequality implies that ek L []. We will deote by K [] the class of covex bodies i R ad by K [] the subclass of bodies of volume. Also, CK [] is the class of covex bodies with ceter of mass at the origi ad SK [] is the class of origi symmetric covex bodies i R. We refer to the books [8], [8] ad [4] for basic facts from the Bru- Mikowski theory ad the asymptotic theory of fiite dimesioal ormed spaces..5 L q cetroid bodies. Let µ P []. For every q ad θ S we defie ( ) /q (.0) h Zq(µ)(θ) := x, θ q f(x) dx, R where f is the desity of µ. If µ L [] the h Zq(µ)(θ) < for every q ad every θ S. We defie the L q -cetroid body Z q (µ) of µ to be the cetrally symmetric covex set with support fuctio h Zq(µ). L q cetroid bodies were itroduced, with a differet ormalizatio, i [] (see also [] where a L q affie isoperimetric iequality was proved). Here we follow the ormalizatio (ad otatio) that appeared i []. The origial defiitio cocered the class of measures K L [] where K is a covex body of volume. I this case, we also write Z q (K) istead of Z q ( K ). 6

7 If K is a compact set i R ad K =, it is easy to check that Z (K) Z p (K) Z q (K) Z (K) for every p q, where Z (K) = cov{k, K}. Note that if T SL the Z p (T (K)) = T (Z p (K)). Moreover, if K is covex body, as a cosequece of the Bru Mikowski iequality (see, for example, []), oe ca check that (.) Z q (K) c 0 q Z (K) for every q ad, more geerally, (.) Z q (K) c 0 q p Z p(k) for all p < q, where c 0 is a absolute costat. Also, if K has its ceter of mass at the origi, the (.3) Z q (K) c K for all q, where c > 0 is a absolute costat. For a proof of this fact ad additioal iformatio o L q cetroid bodies, we refer to [0] ad []..6 Isotropic probability measures. Let µ CP []. We say that µ is isotropic if Z (µ) = B. We write I [] ad IL [] for the classes of isotropic probability measures ad isotropic log-cocave probability measures o R respectively. We say that a covex body K CK [] is isotropic if Z (K) is a multiple of the Euclidea ball. We defie the isotropic costat of K by ( ) / Z (K) (.4) L K := B. So, K is isotropic if ad oly if Z (K) = L K B. We write IK [] for the class of isotropic covex bodies i R. Note that K IK [] if ad oly if L K K L K IL []. A covex body K is called almost isotropic if K has volume oe ad K T (K) where T (K) is a isotropic liear trasformatio of K. We refer to [6], [7] ad [] for additioal iformatio o isotropic covex bodies..7 The bodies K p (µ). A atural way to pass from log-cocave measures to covex bodies was itroduced by K. Ball i []. Here, we will give the defiitio i a somewhat more geeral settig: Let µ P []. For every p > 0 we defie a set K p (µ) as follows: (.5) K p (µ) := { x R : p 0 } f µ (rx)r p dr f µ (0). It is clear that K p (µ) is a star shaped body with gauge fuctio (.6) x Kp(µ) := ( p f µ (0) 0 7 f µ (rx)r p dr) /p.

8 Let k < ad F G,k. For θ S F we defie (.7) θ Bk+ (µ,f ) := θ Kk+ (π F (µ)). I the followig Propositio we give some basic properties of the star-shaped bodies K p (µ). We refer to [], [6], [], [3] for the proofs ad additioal refereces. Propositio.. Let µ P [], p > 0, k < ad F G,k. (i) If µ L [] the K p (µ) K []. Moreover, if µ SL [] the K p (µ) SK []. (ii) If µ CL [] the K + (µ) CK []. If µ SIL [] the K + (µ) SK []. (iii) If µ IL [] the K + (µ) is almost isotropic. (iv) Let p ad µ CL []. The, f µ (0) Z p (µ) Z p ( K + (µ)). (v) Let p k <, F G,k, µ CL [] ad K CL []. The, (.8) f πf (µ)(0) k PF (Z p (µ)) f µ (0) Zp ( B k+ (µ, F )) ad (.9) K F k PF (Z p (K)) Z p ( B k+ (K, F )). (vi) Let k <, F G,k ad K IK []. The, (.0) K F k L B k+ (K,F ) L K. (vii) If µ IL [], the (.) L K+(µ) f µ (0)..8 ψ α -orm. Let µ P []. Give α, the Orlicz orm f ψα of a measurable fuctio f : R R with respect to µ is defied by { (( ) α ) } f(x) (.) f ψα = if t > 0 : exp dµ(x). R t It is ot hard to check that (.3) f ψα sup { f p p /α } : p α. Let θ S. We say that µ satisfies a ψ α -estimate with costat β α,µ,θ i the directio of θ if (.4), y ψα β α,µ,θ, y. 8

9 We say that µ is a ψ α -measure with costat β α,µ where β α,µ := sup θ S β α,µ,θ, provided that this last quatity is fiite. Similarly, if K K [] we defie h Zp(K)(θ) (.5) β α,k := sup sup θ S p α p /α h. Z(K)(θ) Note that β α,µ is a affie ivariat, sice β α,µ T = β α,µ for all T SL. Fially, we defie (.6) P [] (α, β) := {µ P [] : β α,µ β} ad (.7) K [] (α, β) := {K K [] : β α,k β}..9 The parameter k (C). Let C be a symmetric covex body i R. Defie k (C) as the largest positive iteger k for which (.8) µ,k (F G,k : ) W (C)(B F ) P F (C) W (C)(B F ) + k. Thus, k (C) is the maximal dimesio k such that a radom k-dimesioal projectio of C is 4-Euclidea. The parameter k (C) is completely determied by the global parameters W (C) ad R(C): There exist c, c > 0 such that (.9) c W (C) R(C) k (C) c W (C) R(C) for every symmetric covex body C i R. The lower boud appears i Milma s proof of Dvoretzky s theorem (see [3]) ad the upper boud was proved i [9]. 3 Negative momets of the Euclidea orm Let µ P []. If < p, p 0, we defie ( ) /p (3.) I p (µ) := x p dµ(x). R As usual, if K is a Borel subset of R with Lebesgue measure equal to, we write I p (K) := I p ( K ). Defiitio 3.. Let µ P [] ad δ. We defie q (µ) := max{k : k (Z k (µ)) k} q c (µ, δ) := max{p : I p (µ) δ I (µ)} q (µ, δ) := max{k : k (Z k (µ)) k δ }. 9

10 Oe of the mai results of [3] asserts that the momets of the Euclidea orm o log-cocave measures satisfy a strog reverse Hölder iequality up to the value q : Theorem 3.. Let µ CL []. The for every p q (µ), (3.) I p (µ) CI p (µ), where C > 0 is a absolute costat. It is clear from the statemet that i order to apply Theorem 3. i a meaigful way oe should have some o-trivial estimate for the parameter q. The ext propositio (see [, Propositio 3.0] or [3, Propositio 5.7]) gives a lower boud for q, with a depedece o the ψ α costat, i the isotropic case. Propositio 3.3. Let µ I [] P [] (α, β). The (3.3) q (µ) c α β α, where c > 0 is a absolute costat. Defiitio 3.4. Let µ P []. We will say that µ is of small diameter (with costat A > 0) if for every p oe has (3.4) I p (µ) AI (µ). The defiitio that we give here is a direct geeralizatio of the oe give i [] for the case of covex bodies. Let µ P [] ad set B := 4I (µ)b. Note that 3 4 µ(b). We defie a ew measure µ o A i the followig way: for every A A we set µ(a) := µ(a B). µ(b) Assume that, additioally, µ L []. The, it is ot hard to check that (3.5) I (µ) I ( µ), Z (µ) Z ( µ) ad f µ (0) fµ (0). Therefore, if µ L [], we ca always fid a measure µ L [] which is of small diameter (with a absolute costat C > 0) ad satisfies f µ (0) f µ (0). Moreover, if µ is isotropic, the µ is almost isotropic. As a cosequece of [3, Theorem 5.6] we have the followig: Propositio 3.5. Let µ L. The, (3.6) q ( µ, ξ ) q c ( µ, ξ ), where ξ, ξ are absolute costats. 0

11 4 Coheret classes of measures Our startig poit is a simple but crucial observatio from the paper [5] of Bourgai, Klartag ad Milma. First of all, oe may observe that L := sup{l K : K is a covex body i R } is, essetially, a icreasig fuctio of : for every k, L k CL, where C > 0 is a absolute costat. So, usig (.0) we see that if K 0 is a isotropic covex body i R such that L K0 L, the, for all F G,k, (4.) K 0 F /k L B k+ (K 0,F ) L K0 C L k L C. Buildig o the ideas of [5] oe ca use this property of a body K 0 with extremal isotropic costat to get upper bouds for the egative momets of the Euclidea orm o K 0. Sice we wat to apply this argumet i differet situatios, we will first itroduce some termiology. Defiitio 4.. We defie P := i= P []. Similarly, IP := i= IP [], etc. Let U be a subclass of P. Set U [] = U P []. We say that U is coheret if it satisfies the followig two coditios:. If µ U [] the, for all k ad F G,k, π F (µ) U [dim F ].. If m N ad µ i U [i], i =,..., m, the µ µ m U [+ + m]. We also agree that the ull class is coheret. Note that if U ad U are coheret the U U is also coheret. The followig propositio is a traslatio of well kow results to this laguage. Propositio 4.. The classes SP, CP, L, I are coheret. Note that the class K := = {µ P [] : µ = ek ; K K [] } is ot coheret. Propositio 4.3. Let U be a coheret class of measures. If is eve the, for every µ U [], k = ad F G,k, (4.) f πf (µ)(0) k sup f µ (0). µ U [] Moreover, if ρ (U) := sup µ U[] f µ (0), the (4.3) ρ (U) ρ (U) ( ) ρ (U). ρ (U)

12 Proof. For the first assertio use the fact that π F (µ) π F (µ) U [] ad f (πf (µ) π F (µ))(0) = [f πf (µ)(0)]. For the secod assertio use the fact that if µ U [ ] ad µ U [] the we have µ µ U [] ad f µ µ (0) = f µ (0)f µ (0). I particular if a class satisfies e ρ (U) e, it is eough to boud ρ (U) for eve. Note that IL is such a class. I this sectio we itroduce a coheret subclass of ψ a measures, P α (β). Let µ CP []. For every θ S ad every λ > 0 we defie ( ) (4.4) h µ,θ (λ) := h(λ) = log e λ x,θ dµ(x). R Next, if α (, ], we defie (4.5) ψα,µ (θ) := sup λ>0 λ h(λ) α ( ) = sup log e λ x,θ α dµ(x), λ>0 λ R where α is the cojugate expoet of α, i.e. α + α =. Defiitio 4.4. Let µ be a probability measure o R. For α (, ] we defie ψα,µ(θ) (4.6) βµ,α := sup θ S h. Z(µ)(θ) We also defie (4.7) P α (β) := { µ P[] : β µ,α β } = Propositio Let µ CP [], the for every α (, ] ad every θ S we have that (4.8), θ ψα C max{ ψ α,µ (θ), ψ α,µ ( θ)} where C > 0 is a absolute costat.. Let µ SP [], the for every θ S we have that (4.8) C ψ,µ (θ), θ ψ C ψ,µ (θ), where C, C > 0 are absolute costats.

13 Proof. Let α (, ] ad let α [, ) be the cojugate expoet of α. We set ψ := ψ α,µ ( θ), ψ := ψ α,µ (θ), ψ 0 := max{ ψ α,µ (θ), ψ α,µ ( θ)} ad ψ :=, θ ψα. For every λ > 0, (4.9) R e λ x,θ dµ(x) exp(λ α ψ α ). So, by Markov s iequality we get that, for every t > 0, (4.0) µ{x : e λ x,θ e tα e λα ψ α } e t α. Equivaletly, (4.) µ Choosig λ := tα ψ, we get { } x : x, θ tα λ + λα ψ α e tα. (4.) µ{x : x, θ tψ } e tα. Similarly, for every t > 0 we have (4.3) µ{x : x, θ tψ } e tα. Therefore, µ{x : x, θ tψ 0 } = µ{x : x, θ tψ 0 } + µ{x : x, θ tψ 0 } µ{x : x, θ tψ } + µ{x : x, θ tψ } e tα. The last iequality implies that ψ C ψ 0 ad we are fiished with the first part of the propositio. For the secod part we assume that µ is a symmetric ad α =. We oly have to prove the right had iequality i (4.8). Usig that µ is symmetric we have that for every odd k N R x, θ k dµ(x) = 0 So, e λ x,θ dµ(x) = R R λ k x, θ k dµ(x) = k! k=0 (λ) k (k)! (k)k ψ k ) k=0 (eλ ψ) k = exp ( eλ ψ ) k! k=0 k=0 k=0 λ k x, θ k dµ(x) (k)! R (λ) k (k)! (e)k k! ψ k 3

14 It follows that (4.4) ψ := sup λ>0 λ This completes the proof. Corollary 4.6. For every α (, ], ( ) log e λ x,θ dµ(x) e ψ. R (4.5) CP α (β) CP(α, cβ) ad (4.6) SP(, c β) SP (β) SP(, c β), where c, c, c > 0 are uiversal costats. Proof. Ideed, if µ CP α (β) the Propositio 4.5 implies that h ψα(µ)(θ) (4.7) sup θ S h c Z(µ)(θ) sup θ S ψα,µ(θ) cβ h Z(µ)(θ) which meas that µ CP(α, cβ) (recall (.6)). The secod part is proved i a similar way. Next, we prove that the class P α (β) is coheret. The behavior of ψ α,µ for products of measures is described by the followig: Propositio 4.7. Let k be a positive iteger ad let µ i CP [i] ad θ i S i, i =,... k. If ψ α,µi (θ i ) < for all i k ad some α (, ], the (4.8) ψα,µ ((θ,..., θ k )) where µ = µ µ k. ( k i= ψ α α,µ i (θ i ) ) α Proof. For every λ > 0 we ca write ( ) log... e λ P k i= xi,θi dµ k (x k )... dµ (x ) λ α as follows: ( k log λ α i= R i R R k e λ xi,θi dµ i (x i ) ) = 4 λ α λ α k i= k i= k i=, log e λ xi,θi dµ i (x i ) R i α α λ ψ α,µ i (θ i ) ψ α α,µ i (θ i ).

15 Takig the supremum with respect to λ > 0 we get the result. The behavior of margials is described by the followig: Propositio 4.8. Let µ CP []. Let F G,k ad θ S F. If α (, ], the (4.9) ψα,πf (µ)(θ) ψ α,µ (θ). Proof. Note that, for every λ > 0, (4.0) e λ x,θ dµ(x) = R It follows that (4.) λ α F e λ x,θ dπ F (µ)(x) log e λ x,θ dπ F (µ)(x) = F λ log e λ x,θ dµ(x) α R Takig the supremum with respect to λ > 0 we get the result. ψ α α,µ(θ). Propositio 4.9. Let α (, ] ad let β > 0. The the class P α (β) is coheret. Proof. Let µ (P α (β)) []. Fix k < ad F G,k. The, usig (4.8) ad the fact that h Z(π F (µ))(θ) = h Z(µ)(θ) for θ S F, we see that (4.) βπf (µ),α = sup θ S F So, π F (µ) P α (β). ψα,π F (µ)(θ) h Z(π F (µ))(θ) sup θ S F ψα,µ (θ) h Z(µ)(θ) β µ,α. Next, let µ i (P α (β)), i :=,..., k ad set N := [i] + + k. Sice ( k ) h Z(µ µ k )(θ,... θ k ) = i= h Z (θ (µ i) i), we have ψα,µ β µ µ k,α = sup µ k (θ,... θ k ) (θ,...θ k ) S h N Z(µ µ k )(θ,... θ k ) sup (θ,...θ k ) S N β sup (θ,...θ k ) S N β ( k ( k i= ψ α α,µ i (θ i ) i= h Z (µ i) (θ i) ( k ) α ) i= hα Z (θ (µ i) i) ( k i= h Z (θ (µ i) i) sice α [, ), ad x l k α x l k. So, µ µ k P α (β). ) α ) 5

16 5 M-positios ad extremal bodies All the results i this sectio are stated for the case where the dimesio is eve. Propositio 4.3 shows that this is sufficiet for our purposes. However, with mior chages i the proofs, all the results remai valid i the case where the dimesio is odd. Our mai goal i this sectio is to prove the followig: Propositio 5.. Let U IL be a coheret class of probability measures, let ( ) α eve, α (, ) ad t. The, there exists µ U [] such that C 0 α (5.) f µ (0) C sup f ν (0) ν U [] ad (5.) I c ( α)t α (µ ) C 3 t f µ (0), where C 0, C, C 3 > 0 ad c are absolute costats. Moreover, if U = IL, µ ca be chose to be of small diameter (with a absolute costat C 4 > 0). Recall that if K ad C are covex bodies i R, the the coverig umber of K with respect to C is the miimum umber of traslates of C whose uio covers K: { } k (5.3) N(K, C) := mi k N : z,... z k R : K (z i + C). Let K be a covex body of volume i R. Milma (see [4], [5] ad also [6] for the ot ecessarily symmetric case) proved that there exists a ellipsoid E with E =, such that (5.4) log N(K, E) κ, where κ > 0 is a absolute costat. We will use the existece of α-regular M ellipsoids for symmetric covex bodies. More precisely, we eed the followig theorem of Pisier (see [4]; the result is stated ad proved i the case of symmetric covex bodies but it ca be easily exteded to the o-symmetric case): Theorem 5.. Let K be a covex body of volume i R with ceter of mass at the origi. For every α (0, ) there exists a ellipsoid E with E = such that, for every t, (5.5) log N(K, te) κ(α) t α, where κ(α) > 0 is a costat depedig oly o α. Oe ca take κ(α) where κ > 0 is a absolute costat. i= κ α, 6

17 We will also eed the followig facts about ellipsoids: Lemma 5.3. Let E be a ellipsoid i R. Assume that there exists a diagoal matrix T with etries λ λ > 0 such that E = T (B ). The, (5.6) max F G,k E F = ad max P F (E) = ω k F G,k (5.7) mi E F = mi P F (E) = ω k F G,k F G,k for all k. k i= λ i i= k+ Proof. A proof of the equality mi F G,k E F = ω k i= k+ λ i is outlied i [9, Lemma 4.]. Let F s (k) = spa{e k+,..., e }. The, for every F G,k we have (5.8) P Fs(k)(E) = E F s (k) E F P F (E). This shows that (5.9) mi F G,k P F (E) = P Fs(k)(E) = ω k i= k+ ad completes the proof of (5.7). Observe that E = T (B ) is also a ellipsoid; sice the diagoal etries of T are λ λ > 0, the same reasoig shows that ( k ) (5.0) mi E F = mi P F (E ) = ω k λ i. F G,k F G,k Sice P F (E) is a ellipsoid i F ad E F is its polar i F, by the affie ivariace of the product of volumes of a body ad its polar, we get P F (E) E F = B F = ω k for every F G,k. This observatio ad (5.0) prove (5.6). Lemma 5.4. Let be eve ad let E be a ellipsoid i R. Assume that there exists a diagoal matrix T with etries λ λ > 0 such that E = T (B ). The, there exists F G,/ such that P F (E) = λ / (B F ). Proof. The proof ca be foud i [30, pp. 5-6], but we sketch it for the reader s coveiece. { We may assume that λ > > λ > 0. Write = s. The, E e = x R s : } s i= λ i x i (the reaso for this step is that the argumet i [30, pp. 5-6] works i odd dimesios). Sice λ i > λ s > λ s i for every i s, we ca defie b,..., b s > 0 by the equatios (5.) λ i b i + λ s i = λ s(b i + ). i= λ i λ i 7

18 Cosider the subspace F = spa{v,..., v s } G s,s, where v s = e s ad (5.) v i = b ie i + e s i, i =,..., s. b i + It is easy to check that {v,..., v s } is a orthoormal basis for F ad, usig (5.) ad (5.), we see that, for every x F, s (5.3) λ s x = λ s x, v i = i= s i= λ i x, e i = x E. This proves that E F = λs (B F ) ad, by duality, P F (E) = λ s (B F ) = λ / (B F ). Propositio 5.5. Let K ĨK []. Let k ad set (5.4) γ := max F G,k K F k. The, (5.5) mi K H H G, k where 0 < η < is a absolute costat. k γ ( η γ ) k, Proof. Fix α = ad cosider a α-regular M ellipsoid E for K give by Theorem 5.. By the ivariace of the isotropic positio uder orthogoal trasformatios, we may assume that there exists a diagoal matrix T with etries λ λ > 0 such that E = T (B ). Recall that E =. Let F G,k, k. Sice projectig a coverig creates a coverig of the projectio, we have (5.6) P F (K) P F (E) N(K, E) eκ. We will use the Rogers-Shephard iequality (see [7]) for K ad E: sice K =, we kow that (5.7) c ( K F P F (K) ) ( k k ) k e k, where c > 0 is a uiversal costat (see [9] or [7] for the left had side iequality). From (5.7) ad the defiitio of γ i (5.4), we see that (5.8) P F (K) k c γ. 8

19 Usig (5.6) we get (5.9) I other words, c γ e κ k PF (E) k. (5.0) mi P F (E) c k F G,k γ e κ k. We ca ow apply the upper boud from (5.7) to get (5.) c γ E F k κ e k ( PF (E) E F ) k e κ k e k e κ k. It follows that (5.) max H G, k E H eκ γ k c k Lemma 5.3 implies that (5.3) max H G, k P H (E) eκ γ k c k ad hece, (5.4) P H (K) e κ P H (E) eκ γ k c k., for every H G, k, where we have used agai (5.6). agai, we have Applyig (5.7) oce (5.5) c ( K H P H (K) ) k K H This proves that (5.6) mi K H H G, k k γ ( η γ κ k e k ) k ( γ c ) k k. with η = c e κ, as claimed. Lemma 5.6. Let K CK []. Assume that, for some s > 0, (5.7) r s := log N(K, sb ) <. The, (5.8) I rs (K) 3es. 9

20 Proof. Let z 0 R such that K ( z 0 +sb ) K (z +sb ) for every z R. It follows that (5.9) (K + z 0 ) sb N(K, sb ) K =. Let q := r s <. The, usig Markov s iequality, the defiitio of I q (K + z 0 ) ad (5.7), we get (5.30) (K + z 0 ) 3 I q (K + z 0 )B 3 q < e q = e rs From (5.9) we obtai (5.3) (K + z 0 ) 3 I q (K + z 0 )B < (K + z 0 ) sb, ad this implies (5.3) 3 I q (K + z 0 ) s. N(K, sb ). Sice K has ceter of mass at the origi, as a applicatio of Fradelizi s theorem (see [6]), we have that I k (K + z) e I k(k) for ay k < ad z R (a proof appears i [3, Propositio 4.6]). This proves the Lemma. Theorem 5.7. Let be eve ad let K ĨK []. Set (5.33) γ := max F G, K F. The, there exists K ĨK [] such that: (i) η γ L K L K η γl K, where η, η > 0 are absolute costats. (ii) If α (, ) oe has that for every t C γ log N ( K, t B ) C γ κ(α) t α, where κ(α) κ α ad C, C > 0 are absolute costats. (iii) If K is a body of small diameter (with some costat A > ) the K is also a body of small diameter (with costat C 3 γ A >, where C 3 is a absolute costat). Proof. Let E be a α-regular M ellipsoid for K give by Theorem 5.. As i the proof of Propositio 5.5, we assume that E = T (B ) for some diagoal matrix T with etries λ λ > 0. From (5.0) ad Lemma 5.3 we have (5.34) ω ( ) λ ω i= + λ i = mi F G, P F E e κ ( ) c γ, 0

21 ad hece (recall that ω /k k / k), (5.35) λ c. γ Similarly, (5.) ad Lemma 5.3 imply that (5.36) ω ad hece, ( ) λ ω ( ) γ λ i = max E H e κ, H G, c i= (5.37) λ c 3γ. The, by Lemma 5.4 we ca fid F 0 G, (5.38) such that c (B F 0 ) P F0 (E) c 3 γ (B F 0 ). γ Let K 0 := B + (K, F 0 ) ad K := T (K 0 K 0 ) R, where T SL is such that K is isotropic. Note that K 0 K 0 has volume, ceter of mass at the origi ad is almost isotropic. I other words T is almost a isometry. We will show that K satisfies (i), (ii) ad (iii). (i) From Propositio.(vi) we kow that (5.39) c L K K F 0 LK0 c L K K F 0, where c, c > 0 are absolute costats. The, Propositio 5.5 shows that η (5.40) γ L K L K0 η γl K, where η = η c, η = c. Note that L K = L K0. This completes the proof of (i). (ii) From Propositio.(v) ad from the fact that c cov{c, C} Z (C) cov{c, C} for all C i CK [ ], we get ad, similarly, cov{k 0, K 0 } c Z ( B + (K, F 0 )) K F0 PF0 (Z c c (K)) 3 γp F0 (cov{k, K}) c c 3 cov{k 0, K 0 } Z + (K, F 0 )) K F0 PF0 (Z c 4 η c 4 c c 0 γ P F 0 (cov{k, K}),

22 where we have used the fact that Z (K) c 0 Z (K) c c 0 cov{k, K}. I other words, (5.4) c 5 γ P F 0 (cov{k, K}) cov{k 0, K 0 } c 6 γp F0 (cov{k, K}), where c 5, c 6 > 0 are absolute costats. For s > 0 we have N ( K, s B ) = N ( T (K 0 K 0 ), s B ) N(K 0 K 0, cs B ) N(K 0 K 0, cs (B F 0 B F 0 )) N ( K 0, c s B F 0 ), where we have used the fact that T is almost a isometry, ad hece, T (K 0 K 0 ) c (K 0 K 0 ). Moreover, we have used the fact that if K, C are covex bodies, the (5.4) N(K K, C C) N(K, C) ad B k B k B k. Recall that c ad c 3 are the costats i (5.38). For every r > 0, N ( K 0, c 3 rγ (B F 0 ) ) N ( cov{k 0, K 0 }, c 3 rγ (B F 0 ) ) So, we ca write (5.43) N(K, t B ) N N(cov{K 0, K 0 }, rp F0 (E)) N(c 6 γ P F0 (cov{k, K}), rp F0 (E)) N(c 6 γ cov{k, K}, r E) ( ) r N K K, c 6 γ E ( ) r N K, c 6 γ E. ( ) 4 t K, c 7 γ E for every t > 0, where c 7 = c c 6. Sice E is a α-regular ellipsoid for K, for every t c 7 γ we have (5.44) log N(K, t ( ) B t ) 4 log N K, c 7 γ E 4c 7κ(α)γ t α. This completes the proof of (ii).

23 (iii) We have that R(K 0 ) cγa L K. Ideed, by Propositio., ) R (K 0 ) = R ( B + (K, F 0 ) ( ) cr Z + ( B + (K, F 0 ) c K F0 ( R PF0 Z + (K) ) c γr (cov{k, K}) Also, c γr (K) cγa L K. (5.45) R(K ) = R (K 0 K 0 ) = R(K 0 ). To see this, write (5.46) R (K 0 K 0 ) = max (x,y) K 0 K 0 x + y = R (K 0 ). So, usig (i) we get that (5.47) R(K ) R(K 0 ) c γa L K C 3 γ A L K. This completes the proof. Lemma 5.8. Let µ IL []. Fix k < ad F G,k. The, (5.48) K + (µ) F f πf (µ)(0) k k, f µ (0) (5.49) L Bk+ (µ,f ) f πf (µ)(0) k LBk+ ( K +(µ),f ), ad (5.50) f µ (0) B k+ (µ, F ) B k+ ( K + (µ), F ). Proof. We will make use of the followig facts (see Propositio 4. ad Theorem 4.4 i [3]): If µ IL [], the (5.5) f πf (µ)(0) k PF Z k (µ) k, ad if K CK [] the (5.5) K F k PF Z k (K) k. The, takig ito accout Propositio.(iv), we get (5.53) K + (µ) F k PF Z k ( K + (µ)) k fµ (0) PF Z k (µ) f πf (µ)(0) k k. f µ (0) 3

24 This proves (5.48). (ii) Usig Propositio.(v) ad (iv), we have that ( ) )) Z B k+ ( K + (µ), F ) K + (µ) F k PF (Z ( K + (µ) f π F (µ)(0) k f f µ (0) µ (0) PF (Z (µ)) = f πf (µ)(0) k PF (Z (µ)) = f πf (µ)(0) k BF, because Z (µ) = B. Takig volumes we see that (5.54) L Bk+ ( K +(µ),f ) f π F (µ)(0) k ad we coclude by Propositio.(vii) ad (.7). (iii) By Propositio.(v), ( ) (5.55) B k+ (µ, F ) Z k B k+ (µ, F ) π F (µ)(0) k P f µ (0) F Z k (µ) ad, by Propositio.(v) ad the (iv), ( ) B k+ ( K + (µ), F ) Z k B k+ ( K + (µ), F ) )) K + (µ) F k PF (Z k ( K + (µ) We have thus show that π F (µ)(0) k f f µ (0) µ (0) PF (Z k (µ)) = π F (µ)(0) k PF (Z k (µ)). (5.56) B k+ ( K + (µ), F ) π F (µ)(0) k PF (Z k (µ)). Combiig (5.55) ad (5.56) we see that (5.57) f µ (0) B k+ (µ, F ) B k+ ( K + (µ), F ). This completes the proof. Proof of Propositio 5.. (i) Let ν U [] such that sup µ U[] f µ (0) (5.58) K := T ( K + (ν)), = f ν (0). Let 4

25 where T SL is such that K ĨK []. Note that, from Propositio., T is almost a isometry ad L K f ν (0). If U = IL we take K := T ( K + ( ν)). By Propositio. ad (.4) we have that L K f ν (0). The proof of the first two assertios is idetical i both cases. We write µ for either ν or ν. (i) Let F 0 G,, K 0 ĨK [ ] ad K ĨK [] as i the proof of Theorem 5.7. Let µ := π F0 (µ) π F0 (µ). Assume that the two copies of π F0 (µ) live o F ad F respectively, where F G,. Sice µ U ad U is coheret, we have µ U. Moreover, usig agai Propositio., we have that f µ (0) = f πf0 (µ)(0) L K +(π F0 (µ)) = L B +(µ,f 0) L B +( K +(µ),f 0) L B +(K,F 0) = L K0 = L K f µ (0). This settles the first assertio of the Propositio. (ii) Sice U is coheret, for every F G, we have (5.59) f πf (µ)(0) fµ (0). Set γ := max F G, K F. The, (5.60) γ L B +( K +(µ),f ) f π F (µ)(0) L K f µ (0) C, where we have used agai Lemma 5.8. So, by Theorem 5.7 we have that (5.6) log N(K, t B ) C t α ( α). Note that, for every p > 0 ad every pair of probability measures ν, ν livig i F, F respectively, we have P F Z p (ν ν ) = Z p (ν ) ad P F Z p (ν ν ) = Z p (ν ). Ideed, if θ S F, we have that h p Z (θ) = p(ν ν ) x + y, θ p dν (y)dν (x) F F = x, θ p dν (x) = h p Z (θ). p(ν ) Note that for every covex body K ad F G,k oe has (5.6) K P F (K) P F (K). F 5

26 So, we have that ) K + (µ ) P F ( K + (µ ) Therefore, P F (Z ( K + (µ ) ) P F ( K + (µ ) ) P F ( Z ( K + (µ ) ( f µ (0) PF Z (π F 0 (µ) π F0 (µ)) ) ( f µ (0) PF Z (π F 0 (µ) π F0 (µ)) ) f µ (0) Z (π F 0 (µ)) f µ (0) Z (π F 0 (µ)) f µ (0) B + (µ, F 0 ) f µ (0) B + (µ, F 0 ) ) ) B ( K + + (µ), F 0 B ( K + + (µ), F 0 B + (K, F 0 ) B + (K, F 0 ) = K 0 K 0 = K. (5.63) R( K + (µ )) cr(k ) ad (5.64) log N ( K + (µ ), t B ) log N ( K, ct B ) C t α ( α). We have assumed that t α ( a) C, ad hece, by Lemma 5.6 we have (5.65) I p ( K + (µ)) 3et, where p = C t α ( a) <. Note that if µ CL the for every p oe has (see Propositio 3.4 i [3]) (5.66) I p (µ)f µ (0) I p ( K + (µ)). It follows that (5.67) I C (µ ) C t f t α µ (0), ( a) ad the proof of the secod assertio is complete. For the rest of the proof we set µ = ν. I this case, K is a body of small diameter. Ideed, for p, by Propositio.(iv) we have (5.68) I p (K) I p ( K + ( ν)) I p (µ)f ν (0) f ν (0) I ( K + ( ν)) I (K). From Theorem 5.7 we have that K is a body of small diameter, ad this implies. Also, by the first assertio we have that that R(K) I (K ) (5.69) L K f ν (0) fµ (0) LK. ) 6

27 The, from (5.63) we see that for p, (5.70) I p (µ ) I (µ ) I p( K + (µ )) c R( K + (µ )) R(K ) fµ (0) LK I (K ). So, µ is a measure of small diameter. The proof is complete. 6 Proof of the mai result We are ow ready to state ad prove the mai result of the paper: Theorem 6.. Let U be a coheret subclass of IL ad let ad δ. The, ( ) (6.) sup f µ (0) Cδ sup µ U [] µ U [] q c (µ, δ) log e, q c (µ, δ) where C > 0 is a absolute costat. Moreover if U = IL the the supremum o right had side ca be take over all ν IL. Proof. By Propositio 4.3 we ca assume that is eve. Let q := if µ U[] q c (µ, δ). Let α := log (e q ) ad t = C q log e q, where the absolute costat C > 0 ca be chose large eough to esure that t α ( α) C 0, where C 0 > 0 is the costat that appears i Propositio 5.. We have (6.) t α ( α) q log e q log e q = q, ad hece, (6.3) t α ( α) q. By Propositio 5. there exists a measure µ U [] such that f µ (0) sup µ U f µ (0) ad (6.4) I q (µ ) = I c (µ ) C t f t α µ (0) ( α) C q log e fµ (0). q O the other had, by the defiitio of q, we have (6.5) = δ δ I (µ ) I q c(µ,δ)(µ ) I q (µ ). Combiig the above we get the result. 7

28 Remark. Observe that for the choice δ = sup µ U[] f µ (0) we have if q c (µ, δ) µ U [] (see Propositio 4.8 i [3]). This shows that the precedig result is sharp (up to a uiversal costat). Theorem 3. shows that there exists a absolute costat ξ > 0 such that q c (µ, ξ) q (µ) for every µ IL. So we get the followig: Corollary 6.. Let U be a coheret subclass of IL. The for ay, ( ) (6.6) sup f µ (0) e C sup µ U [] µ U [] q (µ) log, q (µ) where C > 0 is a absolute costat. Corollary 6.3. Let α (, ], let β > 0 ad µ (P α (β) IL) []. The, ( (6.7) f µ (0) C α β log α α β ), α where C > 0 is a absolute costat. Proof. Sice µ CP α (β), by Corollary 4.6 we have that µ CP(α, cβ). The, Propositio 3.3 shows that q (µ) c α β. Therefore, the result follows from Corollary 6.. α Theorem 6.4. For every isotropic log-cocave measure µ, (6.8) f µ (0) C 4 log. Moreover, if µ is symmetric ad ψ with costat β > 0, the (6.9) f µ (0) Cβ log β. Proof. (6.8) is a direct cosequace of Corollary 6., Propositio 3.3 ad the fact that every log-cocave measure is ψ with a absolute costat. Recall that, from Corollary 4.6, if µ SP(, β) the µ SP (c β). The (6.9) follows from Corollary 6.3. Remark. I the proof of Corollary 6. we have used the fact that q (µ) q c (µ). Oe may check that i geeral this is ot sharp (for example oe may check that for f µ := fb oe has q (µ) << q c (µ, ξ) for ξ ). As Propositio 3.5 shows, this is ot the case for measures of small diameter. We coclude with the followig: 8

29 Theorem 6.5. The followig statemets are equivalet: (a) There exists C > 0 such that sup (b) There exist C, ξ > 0 such that sup sup f µ (0) C. µ IL [] sup µ IL [] (c) There exists C 3, ξ > 0 such that sup sup µ IL [] q c (µ, ξ ) C. q ( µ, ξ ) C 3. Proof. The claim that (a) implies (b) is a immediate cosequece of the remark after Theorem 6.. The fact that (b) implies (c) follows from Propositio 3.5. Fially from Propositio 3.5 ad theorem 6. we get that (c) implies (a). We close by otig that there is a strog coectio betwee the existece of supergaussia directios ad small ball probability estimates, ad hece, i view of Theorem 6.5, with the hyperplae cojecture as well. This coectio will appear elsewhere. Refereces [] K. M. Ball, Logarithmically cocave fuctios ad sectios of covex sets i R, Studia Math. 88 (988), [] C. Borell, Covex set fuctios i d-space, Period. Math. Hugar. 6, o. (975) 36. [3] J. Bourgai, O the distributio of polyomials o high dimesioal covex sets, Geom. Aspects of Fuct. Aalysis (Lidestrauss-Milma eds.), Lecture Notes i Math. 469 (99), [4] J. Bourgai, O the isotropy costat problem for ψ -bodies, Geom. Aspects of Fuct. Aalysis (Milma-Schechtma eds.), Lecture Notes i Math. 807 (003), 4. [5] J. Bourgai, B. Klartag ad V. D. Milma, Symmetrizatio ad isotropic costats of covex bodies, Geometric Aspects of Fuctioal Aalysis, Lecture Notes i Math. 850 (004), 0 6. [6] M. Fradelizi, Sectios of covex bodies through their cetroid, Arch. Math. 69 (997), [7] A. Giaopoulos, Notes o isotropic covex bodies, Warsaw Uiversity Notes (003). [8] B. Klartag, O covex perturbatios with a bouded isotropic costat, Geom. ad Fuct. Aal. (GAFA) 6 (006)

30 [9] B. Klartag ad V. D. Milma, Rapid Steier Symmetrizatio of most of a covex body ad the slicig problem, Combi. Probab. Comput. 4, o. 5-6 (005) [0] L. Leidler O a certai coverse of Hölders iequality, Liear operators ad approximatio (Proc. Cof., Oberwolfach, 97), Iterat. Ser. Numer. Math., 0, Birkhäuser, Basel, (97), [] E. Lutwak ad G. Zhag, Blaschke-Sataló iequalities, J. Differetial Geom. 47 (997), 6. [] E. Lutwak, D. Yag ad G. Zhag, L p affie isoperimetric iequalities, J. Differetial Geom. 56 (000), 3. [3] V. D. Milma, A ew proof of A. Dvoretzky s theorem i cross-sectios of covex bodies, (Russia), Fukcioal. Aal. i Priloze. 5 (97), o.4, [4] V. D. Milma, Iegalité de Bru-Mikowski iverse et applicatios à la théorie locale des espaces ormés, C.R. Acad. Sci. Paris 30 (986), 5 8. [5] V. D. Milma, Isomorphic symmetrizatio ad geometric iequalities, Geom. Aspects of Fuct. Aalysis (Lidestrauss-Milma eds.), Lecture Notes i Math. 37 (988), [6] V.D. Milma ad A. Pajor, Isotropic positios ad iertia ellipsoids ad zooids of the uit ball of a ormed -dimesioal space, GAFA Semiar 87-89, Spriger Lecture Notes i Math. 376 (989), pp [7] V.D. Milma ad A. Pajor, Etropy ad Asymptotic Geometry of No-Symmetric Covex Bodies, Advaces i Mathematics, 5 (000), [8] V.D. Milma ad G. Schechtma, Asymptotic Theory of Fiite Dimesioal Normed Spaces, Lecture Notes i Math. 00 (986), Spriger, Berli. [9] V.D. Milma ad G. Schechtma, Global versus Local asymptotic theories of fiitedimesioal ormed spaces, Duke Math. Joural 90 (997), [0] G. Paouris, Ψ -estimates for liear fuctioals o zooids, Geom. Aspects of Fuct. Aalysis, Lecture Notes i Math. 807 (003),. [] G. Paouris, O the Ψ -behavior of liear fuctioals o isotropic covex bodies, Studia Math. 68 (005), o. 3, [] G. Paouris, Cocetratio of mass o covex bodies, Geom. Fuct. Aal. 6 (006), [3] G. Paouris, Small ball probability estimates for log cocave measures, Preprit. [4] G. Pisier, The Volume of Covex Bodies ad Baach Space Geometry, Cambridge Tracts i Mathematics 94 (989). [5] A. Prékopa, Logarithmic cocave measures with applicatio to stochastic programmig, Acta Sci. Math. (Szeged), 3 (97), [6] A. Prékopa, A.: O logarithmic cocave measures ad fuctios. Acta Sci. Math. (Szeged), 34 (973), [7] C. A. Rogers ad G. C. Shephard, Covex bodies associated with a give covex body, J. Lodo Soc. 33 (958), [8] R. Scheider, Covex Bodies: The Bru-Mikowski Theory, Ecyclopedia of Mathematics ad its Applicatios 44, Cambridge Uiversity Press, Cambridge (993). 30

31 [9] J. Spigar, A iequality for sectios ad projectios of a covex set, Proc. Amer. Math. Soc. 8 (993), 9 4. [30] C. Zog, Strage pheomea i covex ad discrete geometry, Uiversitext, Spriger (996). Nikos Dafis Departmet of Mathematics Uiversity of Athes Paepistimioupolis 57 84, Athes, Greece ikdafis@googl .com Grigoris Paouris Departmet of Mathematics Texas A & M Uiversity College Statio, TX U.S.A. grigoris paouris@yahoo.co.uk 3