A stability result for mean width of L p -centroid bodies.

Transcription

1 A stability result for mean with of L p -centroi boies. B. Fleury, O. Guéon an G. Paouris Abstract We give a ifferent proof of a recent result of lartag [1] concerning the concentration of the volume of a convex boy within a thin Eucliean shell an proving a conjecture of Anttila, Ball an Perissinaki [1]. It is base on the stuy of the L p -centroi boies. We prove an almost isometric reverse Höler inequality for their mean with an a refine form of a stability result. 1 Introuction In this paper we stuy how the volume of a symmetric convex boy concentrates within a very thin Eucliean shell. Let be an isotropic convex boy in R n i.e. a symmetric convex boy of volume 1 such that for some fixe L > 0, θ S n 1, x, θ x = L. It is known that every symmetric convex boy has an affine image which is isotropic. We enote by x the Eucliean norm of x R n. In the paper [1], Anttila, Ball an Perissinaki aske if every isotropic convex boy in R n satisfy an ε-concentration hypothesis namely: Concentration hypothesis. Does there exist ε n such that lim n ε n = 0 an { x, x 1 L n n} ε ε n? We will prove the following Theorem 1. There exists c an c such that for every isotropic convex boy in R n, an every p log n 1/3, 1/p / 1 x p x x x 1 + c p/log n 1/3. In particular, for every ε 0, 1, { } x, x 1 nl ε e c εlog n 1/1. 1 Université Pierre et Marie Curie, Institut e Mathématiques e Jussieu, boîte 186, 4 Place Jussieu, 755 Paris Ceex 05, FRANCE. fleury bruno@yahoo.fr, gueon@math.jussieu.fr Université e Marne-la-Vallée, Laboratoire Analyse et e Mathématiques Appliquées, 5 B Descartes, Champs-sur- Marne, Marne-la-Vallée Ceex, FRANCE. grigoris paouris@yahoo.co.uk Curie Intra-European Fellowship EIF, Contract MEIF-CT Research supporte by a Marie 1

2 This implies that the concentration hypothesis hols with ε n = clog log n /log n 1/6. This result has been very recently obtaine in full generality by lartag [1], where he prove that 1 hols true with e ε log n for every isotropic convex boy with center of mass at the origin. Our goal is to present a ifferent approach via the notion of L p -centroi boies. To any star shape boy with respect to the origin, L R n, we associate its L p -centroi boy Z p L which is a symmetric convex boy efine by its support function: 1/p y R n, h Zp Ly = x, y x p. L This boy is an homothetic of the L p -centroi boy efine by Lutwak an Zhang in [16] see also [15]. For any symmetric convex boy C, we efine the p-th mean with as 1/p W p C = h C θ p σθ. S n 1 The main result of this paper compares the mean with of the L p -centroi boies of an isotropic convex boy to the mean with of the L p -centroi boies of the Eucliean unit ball of volume 1. Theorem. There exists a constant c such that for any n, for every isotropic convex boy in R n, if D enotes the Eucliean unit ball in R n of volume 1, for every p log n 1/3 W 1 Z p W 1 Z 1 D W 1 Z 1 W 1 Z p D 1 + cp/log n1/3. Regaring as a probability space, these techniques were use by the thir name author [0] to prove that the L q -norms of the Eucliean norm are almost constant for any q n, i.e. see theorem 1. in [0] C 1, q c 1/q 1/ n, x q x C x x = C n L. Theorem 1 is in fact an almost isometric version of this result although it oes not recover the full isomorphic one. It is also relate to a weak form of annan, Lovász an Simonovits [11] conjecture about Cheeger-type isoperimetric constant for convex boies: oes there exist c > 0 such that for any isotropic convex boy, σ := V ar X nl 4 c where X is a ranom vector uniformly istribute on? We refer to the paper of Bobkov [4] for more etails between the full LS-conjecture an this weaker form. Theorem 1 implies that lim n σ /n = 0. Up to now, the only known upper boun was the trivial one, σ c n. On the way, we will nee a new type of stability result for the L p -centroi boies. Let an L be symmetric convex boies of volume 1 in R, if Z p L is close to Z p for the geometric istance, what can we say about the geometric istance between an L? This type of question has been stuie by Bourgain an Linenstrauss [6] in the case of projection boies i.e. p = 1. We will prove a more precise result when one of the boies is the Eucliean unit ball D. The geometric istance between two symmetric convex boies an L is efine by, L = inf {ab a, b > 0 an 1/a L b}.

3 Theorem 3. There exists c > 0 such that for every integer greater than 3 an any o integer p, if is a symmetric convex boy in R such that for some α > 1 an ε 0, c α 3 where = 1/ an D = D 1/ D then, D α an Z p D, Z p 1 + ε, D 1 + hε an 1 hεz p D Z p 1 + hεz p D where hε = c α +p+1 ε 1/. It was prove in [1] that the concentration hypothesis implies some type of central limit theorem. The conjecture about a central limit theorem for convex sets state by Anttila, Ball, Perissinaki [1] an Brehm, Voigt [7] has been recently prove by lartag [1] an we refer to that paper for more precise references on this subject. The paper is organize as follows. In Section, we will explain how we reuce the stuy of concentration of the volume of an isotropic convex boy to the stuy of its L p -centroi boies. We will prove the main Theorem in Section 3. The proof of Theorem 3 is one in Section 4 an uses stanar tools coming from the theory of spherical harmonics. Notations. Throughout this paper, D will be the Eucliean ball in R n an S n 1 the unit sphere. The volume is enote by. We write ω n for the volume of D an σ for the rotationally invariant probability measure on S n 1. Also we write L for the homothetic image of volume 1 of the boy L R n, that is L = L 1/n L an RL will be the circumraius of L i.e. the smallest real number such that L RLD. The letter c will allways be use as being a universal constant an it can change from line to line. Acknowlegement. The authors woul like to thank. Ball for several useful iscussions. Reuction to L p centroi boies. For any isotropic convex boy, we efine I p = x p x 1/p. It is easy to check that there exists a constant c n,p such that for every θ S n 1 c p n,p S n 1 θ, x p σθ = x p, i.e. c n,p = π Γ p+n 1/p Γ p+1 Γ n. Note that c n,p is similar to n + p/p. By Fubini theorem an the efinition of W p Z p, I p = c n,p W p Z p. We first nee some precise computations in the case of the Eucliean ball of volume 1. Lemma 1. Let D be the Eucliean unit ball in R n, then for any p n, I p D/I 1 D 1 + cp/n. 3 Let k be an integer an p k n an enote by D F the Eucliean unit ball of volume 1 in any k-imensional subspace F of R n then W 1 Z 1 D/W 1 Z p D W 1 Z p D F /W 1 Z 1 D F 1 + c p/k. 3

4 Proof. For any 1 p n, we have c n,p W p Z p D 1/p / c n,1 W 1 Z 1 D = x p x x x = 1 + 1/n1 + p/n 1/p 1 + cp/n. ed ed Since for any p 1, W 1 Z p D = W p Z p D an xγx = Γx + 1, we get W 1 Z 1 D W 1 Z p D F W 1 Z p D W 1 Z 1 D F = Γ 1 + n+p Γ 1 + k Γ 1 + n Γ 1 + k+p 1/p Γ 1 + n Γ 1 + k+1 Γ 1 + n+1. Γ 1 + k Simple computations with the Γ function proves the state estimate when p k. For any fixe symmetric convex boy L, Litvak, Milman an Schechtman [14] stuie the behavior of W p L as a function of p. Lemma [14] Let L be a symmetric convex boy of R n then for any p c 1 n W 1 L/RL, p W p L W 1 L h L u W 1 L p c RL 4 n where c 1 an c are universal constants. The next lemma was essentially prove in [0]. Lemma. There exists c > 0 such that for every isotropic convex boy R n, for every 1 p c n, RZ p c p W 1 Z p. 5 Proof. We briefly inicate a proof. In isotropic position, RZ p cprz = cpl. Corollary 3.11 in [0] means that if p c n, W p Z p is similar up to universal constants to W 1 Z p. Observe that W p Z p c p/ni p c pl an p W 1 Z p cpl crz p. Proof of Theorem 1. We write I p I 1 D I 1 I p D = W pz p W 1 Z p From 4 an 5, we get 1 W p Z p /W 1 Z p 1 + c 1 is prove using 3, 6 an Theorem. In particular, W 1 Z p W 1 Z 1 D W 1 Z 1 W 1 Z p D. 6 p n when p c n. Hence Theorem x nl 1 x = I4 4 I 4 1 c/log n1/3. 7 x The function fx = 1 is a polynomial of egree an we can use the results of Bobkov nl [3] about L r -norms of polynomials. Inee, theorem 1 of [3] states that there exists a universal constant c > 0 such that R e ˆfx/c ˆfxx x where ˆf = f 1/. For every ε 0, 1, since ˆfxx f xx 1/4, we get by 7 an by Chebychev inequality { } { x, x 1 nl ε x, x } 1 ε e c ε log n 1/1. 4 nl

5 3 Proof of Theorem We now introuce some notations an recall some well known facts from local theory of Banach spaces. The subspace F R n being chosen, enote by E the orthogonal subspace of F an for every φ S F, the Eucliean sphere in F, we efine Eφ to be {x span{e, φ}, x, φ 0}. For any q 0, efine the star boy B q by its raial function 1/q+1 φ S F, r Bq φ = x, φ x q. Eφ A theorem of Ball [] asserts that when is a symmetric convex boy in R n, this raial function efines a symmetric convex boy in F. These balls are relate with the L p -centroi boies by the following proposition see proposition 4.3 in [0]. Proposition [0] Let be a symmetric convex boy in R n an let 1 k n 1. For every subspace F of R n of imension k an every q 1, we have P F Z q = k + q 1/q Z q B k+q 1 = k + q 1/q B k+q 1 1/k+1/q Z q B k+q 1. 8 Moreover, a simple use of a result of Borell [5] gives comparison between these norms. Lemma [5] For f being a log-concave non-increasing function on [0, +, efine F : t 1 Γt + 0 x t 1 fxx, G : t t then F is log-concave an G is log-convex on 0, x t 1 fxx Proposition 3. Let be a symmetric convex boy in R n, let F be a k-imensional subspace of R n, an for any t 1, efine the symmetric convex boy B t 1 in F as before. For every φ S F an every 1 s t u, we have φ t B t 1 Γs1 λ Γu λ Γt φ 1 λs B s 1 φ uλ B u 1 an φ t B t 1 t s 1 λ u λ φ 1 λs B s 1 φ uλ B u 1 where t = 1 λs + λu. Proof. Let f φ y = E + yφ} for y R + then by Brunn-Minkowski inequality, f φ is a log-concave function an non-increasing. By Fubini, for every φ S F, φ t B t 1 = + 0 y t 1 f φ yy = t 1 Gt = ΓtF t an the conclusion follows easily from the preceing lemma. We will also use the Dvoretzky s theorem prove by Milman [17] see also [18]. Theorem [17] There exist constants c 1, c such that for any n, any ε > 0 an any symmetric convex boy L R n, if k c 1 ε / log1/ε n W 1 L/RL, the set of subspaces F G n,k such that 1 εw 1 LD F P F L 1 + εw 1 LD F where D F is the Eucliean unit ball of F has Haar measure greater than 1 e c k. Remark that it was prove by Goron [9] that we may take ε instea of ε / log1/ε. 5

6 Proof of Theorem. Let be an isotropic convex boy in R n. Hence from 5, for every 1 q c n, RZ q c q W 1 Z q. Without loss of generality, we can assume p to be an o integer. Let k an ε 0, 1/3 to be chosen later be such that k cε n an k p. Since Dvoretzky theorem hols with high probability, we can choose a subspace F of R n of imension k such that five conitions hol simultaneously: for every q {1, p, k, k p, k}, 1 ε W 1Z q W 1 Z q D F Z q D F P F Z q 1 + ε W 1Z q W 1 Z q D F Z q D F. Inee, observe that q {1, p, k, k p, k}, k cε n/q c 1 ε nw 1 Z q /RZ q. From 8, these inclusions mean that for every q {1, p, k,, k p, k}, where 1 εγ q Z q D F Z q B k+q εγ q Z q D F 9 γ q = W 1 Z q k + q 1/q B k+q 1 1/k+1/q W 1 Z q D F. 10 The first step is to prove the following Claim: there is a universal constant c such that, for q {1, p}, B k+q 1, D F c. Inee, since B k+q 1 is a symmetric convex boy in a k-imensional space, it is well known that there exists a universal constant c such that cb k+q 1 Z q B k+q 1 B k+q 1 for q k see for example lemma 4.1 in [19] or lemma in [8]. For q {k, k p, k}, we euce from 9 that B k+q 1, D F c where c is a universal constant. Now, for q {1, p}, Proposition 3 with s = k + q, t = k, u = 3k q i.e. t = 1 λs + λu with λ = 1/ gives φ k B k 1 Γk + q1/ Γ3k q 1/ φ k+q/ B Γk k+q 1 φ 3k q/ B 3k q 1, φ k B k 1 k k + q 1/ φ k+q/ 3k q 1/ B k+q 1 φ 3k q/ B 3k q 1 for every φ S F. Since q p k, it is easy to conclue the proof of the claim. The secon step consists to apply Theorem 3. Inee, for q {1, p}, we get from 9 that Z q B k+q 1, Z q D F 1 + ε/1 ε 1 + 3ε an we have seen that B k+q 1, D F c therefore, Theorem 3 since q is a non even number states that there exists a universal constant c such that 1 h k ε γ q 1 + h k ε 11 an for every θ, θ 0 S F, 1 + h k ε 1 θ 0 Bk+q 1 θ Bk+q h k ε θ 0 Bk+q 1 1 where h k ε = c k 3ε 1/k. We want that this last quantity goes to 0 when k goes to infinity hence we choose ε = c k3 in such a way that h k ε e k. In orer to use Dvoretzky theorem, k has been chosen such that k = cε n which means that k c log n 1/3. By 10 an 11, W 1 Z p W 1 Z 1 D F W 1 Z 1 W 1 Z p D F 1 + e k k + p 1/p B k+p 1 1/k+1/p 1 e k k + 1 B k 1/k. 13 We conclue observing that = 1 can be written as 1 = = kω k x, θ k 1 x σ F θ = kω k S F Eθ 6 S F θ k B k 1 σ F φ

7 so that there exists a θ 0 S F such that 1 = kω k θ 0 k B k 1. Using relation 1, k + p 1/p B k+p 1 1/k+1/p k + 1 B k 1+1/k = k + p 1/p ω k S F θ k ω k S F θ k 1/k+1/p B k+p 1 σ F θ 1+1/k k + 1 B k σ F θ 1 + e k k++k/p k + p 1/p θ 0 k+1 B k k + 1 ω 1 1/p k θ 0 1+k/p B k+p 1. Proposition 3 with s = k, t = k + 1, u = k + p i.e. t = 1 λs + λu with λ = 1/p gives θ 0 k+1 B k Γk1 1/p Γk + p 1/p θ 0 k1 1/p B Γk + 1 k 1 θ 0 1+k/p B k+p 1. Since θ 0 k B k 1 = kω k an p k, simple computations on the Γ function gives k + p 1/p B k+p 1 1/k+1/p k + 1 B k 1+1/k = 1 + e k k 1 + p/k 1/p 1 + e k k 1 k /k Γk + p + 1 Γk + 1 Γk 1 1/p Γk + p 1/p Γk + 1 1/p 1 + cp/k. Combining this last inequality with 13 an with Lemma 1, we get that if p k for a universal constant c. W 1 Z p W 1 Z 1 D 1 + cp/log n1/3 W 1 Z 1 W 1 Z p D 4 Stability result for L p -centroi boies In Theorem 3, the equality case i.e. ε = 0 may be treate via the use of Funck-Hecke theorem. This is why we will follow an approach using the ecomposition in spherical harmonics an we refer to the chapter 3 of the book of Groemer [10] for more etaile explanation. This technique was also use by Bourgain an Linenstrauss [6]. Let p be an o integer with p, we consier the function φ : R R efine by φt = t p an we efine the operator J φ on L S 1 by J φ F u = φ u, v F vσv S 1 for any u S 1. By the Funck-Hecke theorem, for every harmonic polynomial H homogeneous of egree l on the sphere S 1 we have J φ F, H = α,l φ F, H, where, enotes the usual scalar prouct in L S 1 an α,l φ = 1l π 1/ l 1 Γl φt l t l 1 t 3 l+ t. These coefficients are known, see [1] or Lemma 1 in [13]. Hence, for any o values of l, α,l φ = 0 an for any even values of l, α,l φ = π/ 1 Γp + 1 sinπl p/γl p/ p 1. Γl + + p/ 7

8 Stanar computations with the Γ function gives a universal constant c such that for any even integer l, 1 c max, l. 14 α,l φ 1/p+/ For a continuous function F : S 1 R such that F : R R efine by F x = F x/ x is ifferentiable on R \ {0}, we set for any u S 1, 0 F u = F u. The next proposition is a stanar trick using spherical harmonics [10]. Proposition 4. There exists a universal constant c such that for any continuous even function F : S 1 R such that 0 F exists, F c J φ F /+p+ 0 F + F 1 1 /+p+. Proof. Let F Q l F be the ecomposition in spherical harmonics of F with Q l F spherical harmonics of egree l then by Corollary 3..1 in [10] 0 F = l 0 ll + Q l F. For any o l, α,l φ = 0 an since F is even, Q l F = 0. Hence from Parseval equality F = Q l F = α,l φ Q l F β Q l F β α,l φ β where β 0, is chosen such that β/ β = /p + /. By Höler inequality, F α,l φ Q l F By Funck-Hecke theorem, J φ F = c, l β/ Q l F α,l φ β/ β 1 β/. α,l φ Q l F an by inequality 14, Q l F α,l φ /p+/ c max, l Q l F Q l F + l Q l F c F + 0 F., l This proves that F c J φ F /+p+ 0 F + F 1 1 /+p+. We will also nee the following simple lemma. Lemma 5. Let F : S 1 R be a Lipschitz function an let M = max F, F Lip then F 5M 1/+1 F /+1. Proof. Let u S 1 such that F u = F an let Cu, R be the spherical cap of raius R centere at u. For any δ 1, efine A δ = {v S 1, F v δ F } then by Chebychev inequality, σa δ 1 1/δ. For any R 0,, it is well known that σcu, R 1 R 1. If R is chosen such that 1 R 1 = 1 then A δ δ Cu, R. In that case, take v A δ Cu, R then F u F u F v + F v F Lip u v + δ F RM + δ F. 8

9 Since R = /δ 1/ 1, we get the estimate taking δ = M/ F 1/+1 1. Proof of Theorem 3. Using support functions, Z p, Z p D 1 + ε implies that there exists γ > 0 such that γh Zp e D h Zp e 1 + εγh Zp e D. 15 For any symmetric convex boy L R, by integration in polar coorinates, h ZpLu p = x, u p x = ω v, u p 1 σv L + p S 1 hence applying it for L = an L = D, we get for any u S 1, v, u p 1 S 1 v +p γp σv 1 + εp 1 ω 1+p/ v +p L γ p ω 1+p/ where is the norm with unit ball. For every u S 1, let F u = u S 1, S 1 v, u p σv 1, we get S 1 v, u p σv ω1+p/ γ p u p+ 1. Since J φ F J φ F 1 + ε p Since, D α, there exits a, b > 1 such that 1/a D b D an ab = α. y S 1, γ p 1 + ε p h p Z p D e y h p Z p e y x, y p x = h p Z ed/a p D e y/a +p therefore 1/γ p a +p 1 + ε p. For any x R, ω 1/ b 1 x x aω 1/ x an for u S 1, u u, therefore F u = ω1+p/ γ p p+ u p+ u 0 F 0 F p + bp+ γ p 1 + ab 4 α +p ε p. 17 We also have F F 1 + b p+ /γ p α p+ 1 + ε p. Using Proposition 4 with 16 an 17, we get F c1 + ε p 1 /+p+ 6 α +p ε p 1 /+p+ cε /+p+ 4α +p+1. Moreover, for any u, v S 1, F u F v = ω 1+p/ /γ p 1/ u p+ 1/ v p+ an F u F v ω1+p/ +p 1 γ p u v u +p i v i+1 α +p ε p u v. Therefore max F, F Lip 4α +p+1 an by Lemma 5, i=0 F c4α +p+1 ε 4/+1+p+ c4α +p+1 ε 1/ := fε. Recalling the efinition of F, F u = 1 + ω 1+p/ /γ p u p+, u S 1, we have prove 1 fε 1/+p γ p/+p D 1 + fε 1/+p γ p/+p D. 18 Since = D = 1, 1 + fε 1 γ p 1 fε 1 an choosing ε cα 3, 15 an 18 proves the assertions of Theorem 3. 9

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