Geometry of random sections of isotropic convex bodies Apostolos Giannopoulos, Labrini Hioni and Antonis Tsolomitis 6 1 0 2 Abstract p Let K be an isotropic symmetric convex body in Rn. We show that a subspace F Gn,n−k of e ∈ codimension k=γn, where γ (1/√n,1), satisfies S ∈ 8 K∩F ⊆ γc√nLK(B2n∩F) 2 withprobabilitygreaterthan1 exp( √n). Usingadifferentmethodwestudythesamequestionforthe G] Lq-centroidbodiesZq(µ)ofan−isotro−piclog-concaveprobabilitymeasureµon Rn. Forevery16q6n M Wanedaγls∈o g(i0v,e1)bowuensdhsoowntthhaetdaiarmanetdeormofsruabnsdpoamcepFro∈jecGtnio,(n1s−γo)fnZsqa(tµi)sfiaensdZuqs(iµn)g∩thFem⊆wc2e(dγe)d√uqcBe2nth∩atFif. K is an isotropic convex body in Rn then for a random subspace F of dimension (logn)4 one has that . h all directions in F are sub-Gaussian with constant O(log2n). t a m 1 Introduction [ 2 A convex body K in Rn is called isotropic if it has volume K = 1, its center of mass is at the origin (we | | v call these convex bodies “centered”),and its inertia matrix is a multiple of the identity matrix: there exists 4 a constant L >0 such that K 5 2 2 (1.1) hx,θi2dx=L2K 0 ZK 1. for every θ in the Euclidean unit sphere Sn−1. For every centered convex body K in Rn there exists an 0 invertiblelineartransformationT GL(n)suchthatT(K)isisotropic. ThisisotropicimageofK isuniquely 6 determined up to orthogonal tran∈sformations. A well-known problem in asymptotic convex geometry asks 1 if there exists an absolute constant C >0 such that : 1 v i (1.2) Ln :=max LK :K is isotropic in Rn 6C1 X { } r foralln>1(seeSection2forbackgroundinformationonisotropicconvexbodiesandlog-concaveprobability a measures). Bourgainprovedin[5]thatL 6c√4nlogn,andKlartag[19]improvedthis bound toL 6c√4n. n n A second proof of Klartag’s bound appears in [21]. Recall that the inradius r(K) of a convex body K in Rn with 0 int(K) is the largest r > 0 for which ∈ rBn K, while the radius R(K):=max x :x K of K is the smallest R>0 for which K RBn. It is n2ot⊆hard to see that the inradius and t{hke rka2dius∈of a}n isotropic convex body K in Rn satisfy t⊆he bo2unds c L 6 r(K) 6 R(K) 6 c nL , where c ,c > 0 are absolute constants. In fact, Kannan, Lovász and 1 K 2 K 1 2 Simonovits [17] have proved that (1.3) R(K)6(n+1)L . K Radius of random sections of isotropic convex bodies. The first question that we discuss in this articleis to givesharpupper bounds for the radiusofarandom(n k)-dimensionalsectionofK. Anatural − “guess” is that the following question has an affirmative answer. 1 Question 1.1. There exists an absolute constant c > 0 with the following property: for every isotropic 0 convex body K in Rn and for every 16k 6n 1, a random subspace F G satisfies n,n k − ∈ − (1.4) R(K F)6c n/k√nL . 0 K ∩ It was proved in [23] that if K is a symmetric convpex body in Rn then a random F G satisfies n,n k ∈ − (1.5) R(K F)6c(n/k)3/2M˜(K), ∩ where c>0 is an absolute constant and 1 (1.6) M˜(K):= x dx. 2 K k k | |ZK In the case of an isotropic convex body one has K =1 and | | 1/2 (1.7) M˜(K)6 x 2dx =√nL , k k2 K (cid:18)ZK (cid:19) therefore (1.5) implies that a random F G satisfies n,n k ∈ − (1.8) R(K F)6c (n/k)3/2√nL , 1 K ∩ where c >0 is an absolute constant. 1 Our first main result shows that one can have a bound of the order of γ 1√nL when the codimension − K k is greater than γn. Theorem 1.2. Let K be an isotropic symmetric convex body in Rn and let 1 6 k 6 n 1. A random − subspace F G satisfies n,n k ∈ − c n (1.9) R(K F)6 0 √nL K ∩ max k,√n { } with probability greater than 1 exp( √n), where c >0 is an absolute constant. 0 − − The proof is given in Section 3. Note that Theorem 1.2 gives non-trivial information when k >√n. In this case, writing k =γn for some γ (1/√n,1) we see that ∈ c (1.10) R(K F)6 0√nL K ∩ γ with probabilitygreaterthan1 exp( √n) onG . The resultof [23]establishesa γ 3/2-dependence n,(1 γ)n − − − − on γ =k/n. A standard approach to Question 1.1 would have been to combine the low M -estimate with an upper ∗ bound for the mean width (1.11) w(K):= h (x)dσ(x), K ZSn−1 of an isotropic convex body K in Rn, that is, the L -norm of the support function of K with respect to 1 the Haar measure on the sphere. This last problem was open for a number of years. The upper bound w(K) 6 cn3/4L appeared in the Ph.D. Thesis of Hartzoulaki [16]. Other approaches leading to the same K bound can be found in Pivovarov[32] and in Giannopoulos,Paouris and Valettas [15]. Recently, E. Milman showed in [26] that if K is an isotropic symmetric convex body in Rn then (1.12) w(K)6c √n(logn)2L . 3 K 2 In fact, it is not hard to see that his argument can be generalized to give the same estimate in the not necessarily symmetric case. The dependence on n is optimal up to the logarithmic term. From the sharp versionof V. Milman’s low M -estimate (due to Pajorand Tomczak-Jaegermann[28]; see [2, Chapter 7] for ∗ complete references) one has that, for every 16k 6n 1, a subspace F G satisfies n,n k − ∈ − (1.13) R(K F)6c n/kw(K) 4 ∩ p withprobabilitygreaterthan1 exp( c k),wherec ,c >0areabsoluteconstants. Combining (1.13)with 5 4 5 − − E. Milman’s theorem we obtain the folowing estimate: Let K be an isotropic symmetric convex body in Rn. For every 1 6 k 6 n 1, a subspace − F G satisfies n,n k ∈ − c n(logn)2L (1.14) R(K F)6 2 K ∩ √k with probability greater than 1 exp( c k), where c ,c >0 are absolute constants. 3 2 3 − − NotethattheupperboundofTheorem1.2hassomeadvantageswhencomparedto(1.14): Ifkisproportional to n (say k > γn for some γ (1/√n,1)) then Theorem 1.2 guarantees that R(K F) 6 c(γ)√nL for a K random F G . More ge∈nerally, for all k > c6n we have ∩ ∈ n,n−k (logn)4 c n√n c n(logn)2 (1.15) 0 6 2 , max k,√n √k { } and hence the estimate of Theorem 1.2 is stronger than (1.14). Nevertheless, we emphasize that our bound is notoptimal andit wouldbe very interesting to decide whether (1.4) holds true; this wouldbe optimal for all 16k6n. Radius of random sections of L -centroid bodies and their polars. In Section 4 we study the q diameter ofrandomsections ofthe L -centroidbodies Z (µ) ofanisotropiclog-concaveprobabilitymeasure q q µ on Rn. Recall that a measure µ on Rn is called log-concave if µ(λA+(1 λ)B) > µ(A)λµ(B)1 λ for − any compact subsets A and B of Rn and any λ (0,1). A function f : Rn −[0, ) is called log-concave ∈ → ∞ if its support f > 0 is a convex set and the restriction of logf on it is concave. It is known that if a { } probability measure µ is log-concaveand µ(H)<1 for every hyperplane H, then µ is absolutely continuous with respect to the Lebesgue measure and its density f is log-concave; see [4]. Note that if K is a convex µ body in Rn then the Brunn-Minkowskiinequality implies that the indicator function 1 of K is the density K of a log-concavemeasure. We say that a log-concave probability measure µ on Rn is isotropic if its barycenter bar(µ) is at the origin and x,θ 2dµ(x)=1 ZRnh i for all θ Sn 1. Note that the normalization is different from the one in (1.1); in particular, a centered − convex bo∈dy K of volume 1 in Rn is isotropic if and only if the log-concave probability measure µ with K density x Ln1 (x) is isotropic. 7→ K K/LK The L -centroid bodies Z (µ), q >1, are defined through their support function q q 1/q (1.16) h (y):= ,y = x,y qdµ(x) , Zq(µ) kh· ikLq(µ) (cid:18)ZRn|h i| (cid:19) andhaveplayedakeyroleinthestudyofthedistributionoflinearfunctionalswithrespecttothemeasureµ. For every 16q 6n we obtain sharp upper bounds for the radius of random sections of Z (µ) of dimension q proportional to n, thus extending a similar result of Brazitikos and Stavrakakis which was established only for q [1,√n]. ∈ 3 Theorem 1.3. Let µ be an isotropic log-concave probability measure on Rn and let 16q 6n. Then: (i) If k = γn for some γ (0,1), then, with probability greater than 1 e−c4k, a random F Gn,n k satisfies ∈ − ∈ − (1.17) R(Z (µ) F)6c (γ)√q, q 5 ∩ where c is an absolute constant and c (γ)=O(γ 2log5/2(c/γ)) is a positive constant depending only 4 5 − on γ. (ii) With probability greater than 1 e n, a random U O(n) satisfies − − ∈ (1.18) Z (µ) U(Z (µ)) (c √q)Bn, q ∩ q ⊆ 6 2 where c >0 is an absolute constant. 6 The method of proof is based on estimates (from [26] and [11]) for the Gelfand numbers of symmetric convex bodies in terms of their volumetric parameters;combining these generalestimates with fundamental (known)propertiesofthefamilyofthecentroidbodiesZ (µ)ofanisotropiclog-concaveprobabilitymeasure q µweprovideestimatesfortheminimalradiusofak-codimensionalsectionofZ (µ). Then,wepasstobounds q for the radius of random k-codimensionalsections of Z (µ) using known results from [12], [34] and [24]. We q conclude Section 4 with a discussionof the same questions for the polar bodies Z (µ) of the centroidbodies q◦ Z (µ). q Usingthesameapproachwestudythediameterofrandomsectionsofconvexbodieswhichhavemaximal isotropic constant. Set (1.19) L :=max L :K is an isotropic symmetric convex body in Rn . ′n { K } It is known that L 6cL for some absolute constant c>0 (see [9, Chapter 3]). We prove the following: n ′n Theorem 1.4. Assume that K is an isotropic symmetric convex body in Rn with L =L . Then: K ′n (i) A random F G satisfies n,n/2 ∈ (1.20) R(K F)6c √n 7 ∩ and (1.21) L 6c K F 8 ∩ with probability greater than 1 e−c9n, where ci >0 are absolute constants. − (ii) A random U O(n) satisfies ∈ (1.22) K U(K) (c √n)Bn, ∩ ⊆ 10 2 with probability greater than 1 e n, where c >0 is an absolute constant. − 10 − The same arguments work if we assume that K has almost maximal isotropic constant, i.e. L > βL K ′n for some (absolute) constant β (0,1). We can obtain similar results, with the constants c now depending i ∈ only on β. It should be noted that Alonso-Gutiérrez, Bastero, Bernués and Paouris [1] have proved that every convex body K has a section K F of dimension n k with isotropic constant ∩ − n en (1.23) L 6c log . K F ∩ k k r (cid:16) (cid:17) For the proof of this result they considered an α-regular M-position of K. In Theorem 1.4 we consider convex bodies in the isotropic position and the estimates (1.20) and (1.21) hold for a random subspace F. 4 Radius of random projections of L -centroid bodies and sub-Gaussian subspaces of isotropic q convex bodies. Let K be a centered convex body of volume 1 in Rn. We say that a direction θ Sn 1 is − ∈ a ψ -direction (where 16α62) for K with constant b>0 if α (1.24) ,θ 6b ,θ , kh· ikLψα(K) kh· ik2 where (1.25) ,θ :=inf t>0: exp ( x,θ /t)α dx62 . kh· ikLψα(K) |h i| (cid:26) ZK (cid:27) (cid:0) (cid:1) Markov’s inequality implies that if K satisfies a ψ -estimate with constant b in the direction of θ then for α all t>1 we have x K : x,θ >t ,θ 62e ta/bα. Conversely,one cancheck thattail estimates of 2 − |{ ∈ |h i| kh· ik }| this form imply that θ is a ψ -direction for K. α It is well-known that every θ Sn 1 is a ψ -direction for K with an absolute constant C. An open − 1 ∈ question is if there exists an absolute constant C > 0 such that every K has at least one sub-Gaussian direction (ψ -direction) with constant C. It was first proved by Klartag in [20] that for every centered 2 convex body K of volume 1 in Rn there exists θ Sn 1 such that − ∈ t2 (1.26) x K : x,θ >ct ,θ 2 6e−[log(t+1)]2a |{ ∈ |h i| kh· ik }| forallt>1,wherea=3(equivalently, ,θ 6C(logn)a ,θ ). Thisestimatewaslaterimproved kh· ikLψ2(K) kh· ik2 byGiannopoulos,PaourisandValettasin[14]and[15](seealso[13])whoshowedthatthebodyΨ (K)with 2 support function y ,y has volume 7→kh· ikLψ2(K) Ψ (K) 1/n (1.27) c 6 | 2 | 6c logn. 1 2 Z (K) (cid:18)| 2 |(cid:19) p From(1.27)itfollowsthatthereexistsatleastonesub-GaussiandirectionforK withconstantb6C√logn. Brazitikos and Hioni in [7] proved that if K is isotropic then logarithmic bounds for ,θ hold kh· ikLψ2(K) true with probability polynomially close to 1: For any a>1 one has ,θ 6C(logn)3/2max logn,√a L kh· ikLψ2(K) K np o for all θ in a subset Θ of Sn 1 with σ(Θ )>1 n a, where C >0 is an absolute constant. a − a − − Here, we consider the question if one can have an estimate of this type for all directions θ of a subspace F G of dimensionk increasingto infinity with n. We say thatF G is a sub-Gaussiansubspace for n,k n,k ∈ ∈ K with constant b>0 if (1.28) ,θ 6b ,θ kh· ikLψα(K) kh· ik2 forallθ S :=Sn 1 F. InSection5weshowthatifK isisotropicthenarandomsubspaceofdimension F − ∈ ∩ (logn)4 is sub-Gaussian with constant b (logn)2. More precisely, we prove the following. ≃ Theorem 1.5. Let K be an isotropic convex body in Rn. If k (logn)4 then there exists a subset Γ of G n,k ≃ with ν (Γ)>1 n (logn)3 such that n,k − − (1.29) ,θ 6C(logn)2L kh· ikLψ2(K) K for all F Γ and all θ S , where C >0 is an absolute constant. F ∈ ∈ An essential ingredient of the proof is the good estimates on the radius of random projections of the L -centroidbodiesZ (K)ofK,whichfollowfromE.Milman’ssharpboundsontheirmeanwidthw(Z (K)) q q q (see Theorem 5.1). 5 2 Notation and preliminaries We work in Rn, which is equipped with a Euclidean structure , . We denote the correspondingEuclidean h· ·i norm by , and write Bn for the Euclidean unit ball, and Sn 1 for the unit sphere. Volume is denoted k·k2 2 − by . We write ω for the volume of Bn and σ for the rotationally invariantprobability measure on Sn 1. |·| n 2 − We also denote the Haar measure on O(n) by ν. The Grassmann manifold G of k-dimensional subspaces n,k of Rn is equipped with the Haar probability measure ν . Let k 6 n and F G . We will denote the n,k n,k orthogonalprojection from Rn onto F by P . We also define B =Bn F and∈S =Sn 1 F. F F 2 ∩ F − ∩ The letters c,c,c ,c etc. denote absolute positive constants whose value may change from line to line. ′ 1 2 Whenever we write a b, we mean that there exist absolute constants c ,c > 0 such that c a 6 b 6 c a. 1 2 1 2 Also ifA,D Rn wew≃ill write A D ifthere existabsolute constantsc ,c >0suchthat c A D c A. 1 2 1 2 ⊆ ≃ ⊆ ⊆ Convex bodies. A convex body in Rn is a compact convex subset A of Rn with nonempty interior. We say that A is symmetric if A = A. We say that A is centered if the center of mass of A is at the origin, − i.e. x,θ dx=0 for every θ Sn 1. Ah i ∈ − The volume radiusofA is the quantity vrad(A)=(A/Bn )1/n. Integrationinpolarcoordinatesshows R | | | 2| that if the origin is an interior point of A then the volume radius of A can be expressed as 1/n (2.1) vrad(A)= kθk−Andσ(θ) , (cid:18)ZSn−1 (cid:19) where θ = min t > 0 : θ tA . The radial function of A is defined by ρ (θ) = max t > 0 : tθ A , A A k k { ∈ } { ∈ } θ Sn 1. The support function of A is defined by h (y):=max x,y :x A , and the mean width of A − A ∈ h i ∈ is the average (cid:8) (cid:9) (2.2) w(A):= h (θ)dσ(θ) A ZSn−1 ofh onSn 1. The radiusR(A)ofAis the smallestR>0 suchthatA RBn. Fornotationalconvenience we wArite A f−or the homothetic image of volume 1 of a convex body A ⊆Rn, i.2e. A:= A 1/nA. − The polar body A of a convex body A in Rn with 0 int(A) is de⊆fined by | | ◦ ∈ (2.3) A := y Rn : x,y 61for allx A . ◦ ∈ h i ∈ The Blaschke-Santalo´ inequality states t(cid:8)hat if A is centered then A A (cid:9)6 Bn 2, with equality if and only | || ◦| | 2| if A is an ellipsoid. The reverseSantalo´ inequality of J. Bourgainand V. Milman [6] states that there exists an absolute constant c>0 such that (2.4) (A A◦ )1/n >c/n | || | whenever 0 int(A). For ever∈y centered convex body A of volume 1 in Rn and for every q ( n, ) 0 we define ∈ − ∞ \{ } 1/q (2.5) I (A)= x qdx . q k k2 (cid:18)ZA (cid:19) As a consequence of Borell’s lemma (see [9, Chapter 1]) one has (2.6) I (A)6c qI (A) q 1 2 for all q >2. For basic facts from the Brunn-Minkowski theory and the asymptotic theory of convex bodies we refer to the books [33] and [2] respectively. 6 Log-concave probability measures. Let µ be a log-concave probability measure on Rn. The density of µ is denoted by f . We say that µ is centered and we write bar(µ)=0 if, for all θ Sn 1, µ − ∈ (2.7) x,θ dµ(x) = x,θ f (x)dx=0. µ ZRnh i ZRnh i The isotropic constant of µ is defined by 1 (2.8) Lµ := supx∈Rnfµ(x) n [detCov(µ)]21n, f (x)dx (cid:18) Rn µ (cid:19) where Cov(µ) is the covariance matrix of µRwith entries x x f (x)dx x f (x)dx x f (x)dx (2.9) Cov(µ) := Rn i j µ Rn i µ Rn j µ . ij f (x)dx − f (x)dx f (x)dx R Rn µ R Rn µ R Rn µ We say that a log-concaveprobabilityRmeasure µ on RnRis isotropic if bRar(µ)=0 and Cov(µ) is the identity matrix. Note that a centered convex body K of volume 1 in Rn is isotropic,i.e. it satisfies (1.1), if and only if the log-concave probability measure µ with density x Ln1 (x) is isotropic. Note that for every log-concavemeasure µ on Rn one has K 7→ K K/LK (2.10) L 6κL , µ n where κ>0 is an absolute constant (a proof can be found in [9, Proposition 2.5.12]). We will use the following sharp result on the growth of I (K), where K is an isotropic convex body in q Rn, proved by Paouris in [29] and [30]. Theorem 2.1 (Paouris). There exists an absolute constant δ > 0 with the following property: if K is an isotropic convex body in Rn, then 1 1 (2.11) √nL = I (K)6I (K)6I (K)6δI (K)=δ√nL K 2 q q 2 K δ δ − for every 16q 6√n. For every q >1 and every y Rn we set ∈ 1/q (2.12) h (y)= x,y qdµ(x) . Zq(µ) (cid:18)ZRn|h i| (cid:19) The L -centroid body Z (µ) of µ is the symmetric convex body with support function h . Note that µ q q Zq(µ) is isotropic if and only if it is centered and Z (µ) = Bn. If K is an isotropic convex body in Rn we define 2 2 Z (K) = L Z (µ ). From Ho¨lder’s inequality it follows that Z (K) Z (K) Z (K) Z (K) for all q K q K 1 p q 16p6q 6 , where Z (K)=conv K, K . Using Borell’s lemma,⊆one can ch⊆eck that⊆ ∞ ∞ ∞ { − } q (2.13) Z (K) c Z (K) q 1 p ⊆ p for all 16p n. q 2 It was shown by P⊇aouris∞[29] that if 16q 6√n then (2.14) w Z (µ) √q, q ≃ and that for all 16q 6n, (cid:0) (cid:1) (2.15) vrad(Z (µ))6c √q. q 1 7 Conversely, it was shown by B. Klartag and E. Milman in [21] that if 16q 6√n then (2.16) vrad(Z (µ))>c √q. q 2 This determines the volume radius of Z (µ) for all 1 6 q 6√n. For larger values of q one can still use the q lower bound: (2.17) vrad(Zq(µ))>c2√qL−µ1, obtained by Lutwak, Yang and Zhang in [25] for convex bodies and extended by Paouris and Pivovarov in [31] to the class of log-concave probability measures. Let µ be a probability measure on Rn with density f with respect to the Lebesgue measure. For every µ 16k 6n 1 and every E G , the marginalof µ with respect to E is the probability measure π (µ) on n,k E − ∈ E, with density (2.18) f (x)= f (y)dy. πE(µ) µ Zx+E⊥ It is easily checked that if µ is centered, isotropic or log-concave, then π (µ) is also centered, isotropic or E log-concave,respectively. A very useful observation is that: (2.19) P Z (µ) =Z π (µ) F q q F for every 16k6n 1 and every F G (cid:0) . (cid:1) (cid:0) (cid:1) n,n k If µ is a centered−log-concavepro∈bability−measure on Rn then for every p>0 we define (2.20) Kp(µ):=Kp(fµ)= x: ∞rp−1fµ(rx)dr > fµ(0) . p (cid:26) Z0 (cid:27) From the definition it follows that K (µ) is a star body with radial function p (2.21) ρKp(µ)(x)= f 1(0) ∞prp−1fµ(rx)dr 1/p (cid:18) µ Z0 (cid:19) for x=0. The bodies K (µ) were introduced in [3] by K. Ball who showedthat if µ is log-concavethen, for p 6 every p>0, K (µ) is a convex body. p If K is isotropic then for every 16k 6n 1 and F G , the body K (π (µ )) satisfies − ∈ n,n−k k+1 F⊥ K L (2.22) K F 1/k Kk+1(πF⊥(µK)). | ∩ | ≃ L K For more information on isotropic convex bodies and log-concavemeasures see [9]. 3 Random sections of isotropic convex bodies TheproofofTheorem1.2isbasedonLemma3.1andLemma3.2below. TheyexploitsomeideasofKlartag from [18]. Lemma 3.1. Let K be an isotropic convex body in Rn. For every 1 6 k 6 n 1 there exists a subset − := (n,k) of G with ν ( )>1 e √n that has the following property: for every F , n,n k n,n k − A A − − A − ∈A (3.1) x K F : x >c √nL 6e (k+√n) K F , 2 1 K − |{ ∈ ∩ k k }| | ∩ | where c >0 is an absolute constant. 1 8 Proof. Integration in polar coordinates shows that for all q >0 (n k)ω (n k)ω (3.2) ZGn,n−kZK∩F kxkk2+qdxdνn,n−k(F)= −nωnn−k ZKkxkq2dx= −nωnn−kIqq(K), and an application of Markov’s inequality shows that a random F G satisfies n,n k ∈ − (n k)ω (3.3) kxkk2+qdx6 −nω n−k(eIq(K))q ZK∩F n with probability greater than 1 e q. − − Fix a subspace F G which satisfies (3.3). From (2.22) we have n,n k ∈ − (3.4) K F 1/k >c2LKk+1(πF⊥(µK)) > c3 | ∩ | L L K K where c ,c >0 are absolute constants. A simple computation shows that 2 3 (n k)ω (3.5) − n−k 6(c4√n)k nω n for an absolute constant c >0. Using also (2.11) with q =√n we get 4 1 1 (n k)ω (3.6) K F kxkk2+√ndx6 K F −nω n−k(eI√n(K))√n | ∩ |ZK∩F | ∩ | n 6(c L )k(c √n)k(eδ√nL )√n 6(c √nL )k+√n, 5 K 4 K 6 K where c >0 is an absolute constant. It follows that 6 (3.7) x K F : x >ec √nL 6e (k+√n) K F . 2 6 K − |{ ∈ ∩ k k }| | ∩ | and the lemma is proved with c =ec . ✷ 1 6 The next lemma comes from [18]. Lemma 3.2 (Klartag). Let A be a symmetric convex body in Rm. Then, for any 0<ε<1 we have 1 (3.8) x A: x >εR(A) > (1 ε)m A. 2 |{ ∈ k k }| 2 − | | Proof. Let x A such that x =R(A) and define v =x / x . We consider the set A+ defined as 0 0 2 0 0 2 ∈ k k k k (3.9) A+ := x A: x,v >0 . { ∈ h i } Since A is symmetric, we have A+ = A/2. Note that | | | | (3.10) x A: x >εR(A) εx +(1 ε)A+. 2 0 { ∈ k k }⊇ − Therefore, 1 (3.11) x A: x >εR(A) > εx +(1 ε)A+ =(1 ε)m A+ = (1 ε)m A, 2 0 |{ ∈ k k }| | − | − | | 2 − | | as claimed. ✷ Proof of Theorem 1.2. Let K be an isotropic symmetric convex body in Rn. Applying Lemma 3.1 we find a subset of G with ν ( )>1 e √n such that, for every F , n,n k n,n k − A − − A − ∈A (3.12) x K F : x >c √nL 6e (k+√n) K F . 2 1 K − |{ ∈ ∩ k k }| | ∩ | 9 We distinguish two cases: Case 1. If k >n/3 then choosing ε0 =1 e−31 we get − 1 1 (3.13) (1 ε0)n−k K F = e−n−3k K F >e−n−3k−1 K F >e−(k+√n) K F , 2 − | ∩ | 2 | ∩ | | ∩ | | ∩ | because k+√n> n k +1. By Lemma 3.2 and (3.12) we get that −3 (3.14) x K F : x >ε R(K F) > x K F : x >c √nL , 2 0 2 1 K |{ ∈ ∩ k k ∩ }| |{ ∈ ∩ k k }| therefore (3.15) R(K F)0 is an absolute constant. Case 2. If k 6 n/3 then we choose ε = k+√n. Note that ε < 1/2. Using the inequality 1 t > e 2t on 1 6(n k) 1 − − − (0,1/2) we get n k (3.16) 1(1 ε1)n−k K F = 1 1 k+√n − K F >e−k+3√n−1 K F >e−(k+√n) K F , 2 − | ∩ | 2 − 6(n k) | ∩ | | ∩ | | ∩ | (cid:18) − (cid:19) because 2(k+√n) >1. By Lemma 3.2 this implies that 3 (3.17) x K F : x >ε R(K F) > x K F : x >c √nL , 2 1 2 1 K |{ ∈ ∩ k k ∩ }| |{ ∈ ∩ k k }| therefore (3.18) ε R(K F) 0. This completes the proof of the theorem (with a probability estimate 3 1 e √n for all 16k6n 1). ✷ − − − Remark 3.3. It is possible to improve the probability estimate 1 e √n in the range k > γn, for any − − γ (1/√n,1). Thiscanbe donewiththe helpofknownresultsthatdemonstratethe factthattheexistence ∈ of one s-dimensional section with radius r implies that random m-dimensional sections, where m0 is an absolute constant. − − 2 − We applythis resultasfollows. Letk =γn>√nandsett=δn,whereδ γ/log(1+1/γ). Fromthe proof ≃ of Theorem 1.2 we know that there exists E G such that n,n t ∈ − c n (3.21) R(K E)6 1 √nL , K ∩ t 10

Geometry of random sections of isotropic convex bodies

Similar Geometry of random sections of isotropic convex bodies