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On the volume of caps and bounding the mean-width of an isotropic convex body Accepted for publication in Math. Proc. Cambridge Phil. Soc. Peter Pivovarov∗ March 12th, 2010 Abstract Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width. 1 Introduction Let K be a convex body in Rn with volume vol (K) = 1 and suppose its center of mass is the origin. As is commonly done in the asymptotic theory of convex bodies, we treat K as a probability space. In particular, for each unit vector θ, we view the linear functional h·, θi : K → R given by hx, θi = x1 θ1 + . . . + xn θn , (x ∈ K), as a random variable on K. Motivated by Bourgain’s approach to the Slicing Problem [4], recent research has focused on the distribution of the functionals h·, θi. In particular, efforts have been made to show that for any such K, there exists a direction θ which exhibits sub-Gaussian tail-decay, meaning that, 2 vol {x ∈ K : |hx, θi| > t kh·, θik2 } ≤ e−ct (1) 2 for R all t 2≥ 1, where c > 0 is an absolute constant and kh·, θik2 = |hx, θi| dx. The papers [7] and [15] contain the latest developments, as K well as further motivation and history. In this article, we consider bounds ∗ This article is part of the author’s Ph.D. thesis, written under the supervision of Professor Nicole Tomczak-Jaegermann at the University of Alberta. The author holds an Izaak Walton Killam Memorial Scholarship. 1 involving the reverse inequality, namely super-Gaussian estimates of the form 2 vol {x ∈ K : |hx, θi| > t kh·, θik2 } ≥ e−ct , (2) for t > 0 in some suitable range. Such estimates are garnering increased attention, as in [14] and [21], and are the starting point for this paper. Our first main result concerns super-Gaussian directions for convex bodies that are isotropic and 1-unconditional. By isotropic, we mean that K has volume one, center of mass at the origin and each functional has the same variance, i.e., Z hx, θi2 dx = L2K for each θ ∈ S n−1 , (3) K where LK is the isotropic constant of K. Any convex body has an affine image which is isotropic and LK is an affine-invariant (see, e.g., the survey [19]). By 1-unconditional, we mean that for each x = (x1 , . . . , xn ) ∈ K, the parallelepiped [− |x1 | , |x1 |] × . . . × [− |xn | , |xn |] is contained in K. For such bodies, there are many super-Gaussian directions, which we gauge in terms of the Haar measure σ on the sphere S n−1 . Proposition 1.1. There exists an absolute constant C ≥ 1 such that for any integer n ≥ 1, and any 1-unconditional isotropic convex body K ⊂ Rn , the σ-measure of the set of θ ∈ S n−1 such that vol ({x ∈ K : |hx, θi| ≥ t}) ≥ exp(−Ct2 ) whenever C≤t≤ √ n , C log n (4) (5) is at least 1 − 2−n . The isotropic constant LK = kh·, θik √ from the √ 2 has been omitted statement since such bodies satisfy 1/ 2πe ≤ LK ≤ 1/ 2 (see, e.g., [3]). Proposition 1.1 complements results of Bobkov and Nazarov [2], who treated the case of sub-Gaussian directions. As a corollary, we get lower bounds for the volume of caps defined in terms of the width of the body, measured in terms of the support function hK (θ) := supx∈K hx, θi. e = C(β) e Corollary 1.2. Let β > 0. Then there is a constant C such that for any integer n ≥ 1, and any 1-unconditional isotropic convex body K ⊂ Rn , the σ-measure of the set of θ ∈ S n−1 satisfying whenever e 2 n log n) vol ({x ∈ K : |hx, θi| ≥ εhK (θ)}) ≥ exp(−C Cε q C e log n Cn ≤ε≤ C p 1 e log3/2 n C (6) (7) e = 3(β + 1) and C is the constant from is at least 1 − 2−n − 2n−β , where C Proposition 1.1. 2 For us, the motivation for bounding the volume of the caps in (6) comes from a paper by Giannopoulos and Milman [10] involving approximation of a convex body by a random polytope. Corollary 1.2 shows that for 1-unconditional bodies, one has better estimates in most directions. Such estimates, in turn, are intimately related to mean-width, which leads us to the second purpose of this paper. Finding the correct upper bound for the mean-width of an arbitrary isotropic convex body (not necessarily 1-unconditional) is a problem wellknown to specialists. It has numerous connections and implications (some of which we review below). We connect the latter problem with volume estimates for caps, similar to (6), and give a sufficient condition under which one can bound the mean-width. Our approach may be of independent interest since it involves approximating a convex body by a random polytope with relatively few vertices. The paper is organized as follows. The proofs of Proposition 1.1 and Corollary 1.2 are in §2, the first three subsections of which point out the key ingredients. The observations about mean-width are contained in §3, the main result being Proposition 3.4. Lastly, a few words on notation and viewpoint. Our results are most meaningful when the dimension n is large. Throughout, c, c1 , C, C ′ , . . ., etc. denote absolute constants (in particular, independent of n and K). The symbol |·| will serve the dual role of the standard Euclidean norm on Rn and also the absolute value of a scalar, the use of P which will be clear from the context; for x = (x1 , . . . , xn ) ∈ Rn , kxk1 = n i=1 |xi | and kxk∞ = maxi≤n |xi |. 2 Super-Gaussian estimates in 1-unconditional isotropic convex bodies We begin by isolating the key ingredients in the proof of Proposition 1.1. Our proof generalizes an argument due to Schmuckenschläger [27, Proposition 3.4], who showed that the diagonal direction θd = √1n (1, 1, . . . , 1) is super-Gaussian for the unit ball of ℓn p for 1 ≤ p ≤ ∞. Our first step is to pass from θd to a large subset of directions with “well-spread” coordinates. 2.1 Well-spread vectors on the sphere The next lemma identifies the set of directions for which we will establish estimate (4) in Proposition 1.1. Similar facts have been used in various problems (e.g., the use of “incompressible” vectors as in [26]). We include a proof for completeness. Throughout, we use the following notation [n] := {1, . . . , n}. (8) Lemma 2.1. There exist absolute constants C1 > 0 and κ > 0 such that 3 for any integer n ≥ 1, the set Θ := θ ∈ S n−1 |∃I = I(θ) ⊂ [n] with #I ≥ κn : 1 √ C1 C1 ≤ |θi | ≤ √ ∀ i ∈ I n n (9) has σ-measure at least 1 − 2−n . For the proof, we will use the following standard facts. Lemma 2.2. There exists an absolute constant c′ > 0 such that for any integer n ≥ 1, the set √ √ Θ′ := {θ ∈ S n−1 : c′ n ≤ kθk1 ≤ n} (10) has σ-measure at least 1 − 2−n . Lemma 2.2 follows from, e.g., [20, §2.3 & §5.3] or [25, Theorem 6.1]. The second fact we need is the Paley-Zygmund inequality. Lemma 2.3. Let Z be a non-negative random variable with finite variance. Then for each t ∈ (0, 1), P (Z ≥ tEZ) ≥ (1 − t)2 (EZ)2 . EZ 2 (11) For a proof, see, e.g., [16, Lemma 0.2.1 ]. Proof of Lemma 2.1. Fix θ ∈ Θ′ (from Lemma 2.2) and write αi := √ |θi | n. Without loss of generality, we may assume that the αi are distinct. Let Z be a random variable such that P (Z = αi ) = 1/n. Then EZ = n n 1X 1 X |θi | αi = √ n i=1 n i=1 and hence c′ ≤ EZ ≤ 1. By Markov’s inequality, for any λ > 0, we have P (Z > λ) ≤ P (Z > λEZ) ≤ 1 λ and hence we obtain #{i ∈ [n] : αi ≤ λ} ≥ (1 − λ−1 )n. (12) Next, observe that EZ 2 = n n 1X 2 X 2 αi = θi = 1. n i=1 i=1 By the Paley-Zygmund inequality (Lemma 2.3), we have P Z ≥ c′ /2 ≥ P (Z ≥ (1/2)EZ) ≥ (c′ )2 /4. and therefore #{i ∈ [n] : αi ≥ c′ /2} ≥ (c′ )2 n/4. By (12) and (13), we conclude the result. 4 (13) 2.2 Main probabilistic ingredients The proof of Proposition 1.1 relies on two theorems about independent Rademacher random variables ε1 , . . . , εn , i.e., P (εi = 1) = P (εi = −1) = 1/2, i = 1, . . . , n, (14) which we state here for the reader’s convenience. The first is the Contraction Principle; see, e.g., [17, Theorem 4.4]. Theorem 2.4. Let ε1 , . . . , εn be independent Rademacher random variables. Let x1 , . . . , xn be elements of a Banach space B and let α1 , . . . , αn be real numbers such that |αi | ≤ 1 for all i = 1, . . . , n. Then for any t > 0, n n ! ! X X P αi εi xi > t ≤ 2P εi xi > t . (15) i=1 i=1 The second ingredient is the following theorem about super-Gaussian estimates for Rademacher sums, which can be found in [17, §4.1]. Theorem 2.5. There is an absolute constant C2 ≥ 1 such that if ε1 , . . . , εn are independent Rademacher random variables (as in (14)) and if s ∈ R and ξ ∈ Rn satisfy |ξ|2 C2 |ξ| ≤ s ≤ , (16) C2 kξk∞ then ! n X P εi ξi ≥ s ≥ exp(−C2 s2 / |ξ|2 ). (17) i=1 To show that each θ ∈ Θ (Lemma 2.1) satisfies the super-Gaussian estimate (4), Theorem 2.4 will be used to pass to subspaces E(I) := span{ei : i ∈ I} on which we have control of the coordinates of θ. To use Theorem 2.5, we will need volume estimates for certain sets involving the |·| and k·k∞ norms on the orthogonal projection of K onto E(I). This is done in the next section. 2.3 Projections and retention of volume Here we prove a lemma which gives a uniform lower bound for the volume of certain sets that will be used in conjunction with Theorems 2.4 and 2.5. We emphasize that it is a general fact, true for arbitrary isotropic convex bodies and arbitrary subspaces (not just unconditional bodies and coordinate subspaces as we need here). For 1 ≤ ℓ ≤ n, let Gn,ℓ denote the set of all ℓ-dimensional subspaces of Rn ; for E ∈ Gn,ℓ , let PE be the orthogonal projection onto E. Lemma 2.6. There exist positive absolute constants C ′ , C ′′ and c such that for each integer n ≥ 1, for any isotropic convex body K ⊂ Rn , any ℓ ∈ [n] and any E ∈ Gn,ℓ , the intersection of the sets √ √ ′ KE := {x ∈ K : (1/C ′ ) ℓLK ≤ |PE x| ≤ C ′ ℓLK } (18) and say KE := ′ KE ′′ KE := {x ∈ K : kPE xk∞ ≤ C ′′ LK log n}, ∩ ′′ , KE has volume greater than c. 5 (19) The proof relies on two basic facts. See, for instance, [8, Proposition 2.5.1] and [8, Proposition 2.1.1]. Fact 2.7. There exists an absolute constant C3 such that if n ≥ 1, K ⊂ Rn is an isotropic convex body and N is a finite subset of the Euclidean ball B2n , then Z max |hx, θi| dx ≤ C3 LK log(#N ). K θ∈N Fact 2.8. There exists an absolute constant C4 such that if n ≥ 1, K ⊂ Rn is a convex body of volume one, and if f : Rn → R is a semi-norm, then Z f p (x)dx K 1/p ≤ C4 p Z f (x)dx K for all p ≥ 1. (20) Proof of Lemma 2.6. Let K ⊂ Rn be an isotropic convex body, ℓ ∈ [n] and E ∈ Gn,ℓ . Then Z K |PE x| dx ≤ Z K |PE x|2 dx 1/2 = √ ℓLK . By Markov’s inequality, for any τ > 0, we have √ 1 vol {x ∈ K : |PE x| ≤ τ ℓLK } ≥ 1 − 2 . τ (21) Setting c1 := (2C4 )−1 , where C4 is the constant from Fact 2.8, we have Z K |PE x| dx ≥ c1 Z K |PE x|2 dx 1/2 √ = c1 ℓLK . Applying the Paley-Zygmund inequality (Lemma 2.3), we get √ vol {x ∈ K : |PE x| ≥ (c1 /2) ℓLK } ≥ c21 /4. (22) Taking into account (21) and (22), we determine that there are positive absolute constants C ′ and c > 0 for which √ √ ′ vol KE = {x ∈ K : (1/C ′ ) ℓLK ≤ |PE x| ≤ C ′ ℓLK } ≥ 2c. (23) To conclude, set C ′′ := C3 /c, where C3 is the constant from Fact 2.7. Since kPE xk∞ = max |hPE x, ei i| = max |hx, PE ei i| , i≤n i≤n we can apply Markov’s inequality and Fact 2.7 to obtain ′′ vol KE = vol x ∈ K : kPE xk∞ ≤ C ′′ LK log n ≥ 1 − c. Thus ′ ′′ vol (KE ) = vol KE ∩ KE ≥ c, which concludes the proof. 6 2.4 Proof of Proposition 1.1 Here we combine the results of the previous sections to complete the proof. Proof of Proposition 1.1. Assume K is a 1-unconditional isotropic convex body in Rn . Consider C1 , κ and Θ from Lemma 2.1. Set ℓ := ⌊κn⌋, the largest integer less than κn. Fix θ ∈ Θ so that C1 1 √ ≤ |θi | ≤ √ C1 n n for all i ∈ I, where I = I(θ) ⊂ [n] and |I| = ℓ. Set E(I) := span{ei : i ∈ I}, n where the ei ’s are the standard unit vector basis for R . By Lemma 2.6, ′ ′′ the intersection KE(I) ∩ KE(I) has volume vol KE(I) ≥ c. Let X = (x1 , . . . , xn ) be a random vector distributed uniformly in K. Let ε1 , . . . , εn be independent Rademacher random variables (cf. (14)). Then X and (ε1 X1 , . . . , εn Xn ) have the same distribution. Denote the probability measure corresponding to X, namely vol (·|K ), by PK ; by Pε the product-measure corresponding to ε = (ε1 , . . . , εn ). Then n ! X vol ({x ∈ K : |hx, θi| > t}) = PK θi xi > t i=1 n ! X = PK ⊗ Pε εi θi xi > t i=1 n ! Z X = Pε εi θi xi > t dx K i=1 ! Z X ≥ (1/2) Pε εi θi xi > t dx (by Thm. 2.4) K i∈I ! Z X ≥ (1/2) Pε εi θi xi > t dx. (24) KE(I) i∈I Fix x ∈ KE(I) , and set y = (θi xi )i∈I . Then, by definition of KE(I) and Θ, C1 C ′′ LK log n C1 √ kyk∞ ≤ √ PE(I) x∞ ≤ n n and √ √ 1 C1 C1 C ′ ℓLK ℓLK √ √ √ √ ≤ . P x ≤ |y| ≤ P x ≤ E(I) E(I) C1 C ′ n C1 n n n √ Since K is 1-unconditional, LK ≤ √ 1/ 2, (e.g., [3]). Moreover, for any convex body K, LK ≥ LB2n ≥ 1/ 2πe. Recalling that ℓ = ⌊κn⌋, we conclude that there exist absolute constants A1 > 1 and A2 > 1 such that kyk∞ ≤ A1 log n √ n 7 and 1 ≤ |y| ≤ A2 . A2 Let C2 be the constant from Theorem 2.5. At this point we can determine the constant C asserted in Proposition 1.1: take C := A1 A22 C2 . Then our assumption (5) implies C2 |y| ≤ t ≤ |y|2 , C2 kyk∞ making (24) ripe for an application of Theorem 2.5: ! Z Z X (1/2) Pε εi θi xi > t dx ≥ (1/2) exp(−C2 t2 / |y|2 )dx KE(I) KE(I) i∈I ≥ (c/2) exp(−C2 A22 t2 ) ≥ (c/2) exp(−Ct2 ), 2 where we have used the notation y = (θi xi )i∈I as above. Since c/2 ≥ e−t for t large enough, we can recover the proposition as stated simply by adjusting the constants. Remark 2.9. The idea in the proof of Proposition 1.1 was recently adapted and used in [5, Lemma 2.7]. 2.5 Proof of Corollary 1.2 To prove Corollary 1.2, we will need two additional results. Lemma 2.10. For any M ∈ (0, 1), the set Θ1 := {θ ∈ S n−1 : kθk∞ ≤ M } has σ-measure at least 1 − 2ne−nM 2 /2 . Proof. Using the well-known estimate σ(θ ∈ S n−1 : |he1 , θi| > M ) ≤ 2e−nM 2 /2 , (25) (see, e.g., [1, Lemma 2.2]), we have σ θ ∈ S n−1 : ∃i ≤ n : |hei , θi| > M ≤ nσ θ ∈ S n−1 : |he1 , θi| > M ≤ 2ne−nM 2 /2 . Another result, due to Bobkov and Nazarov [3, Propositions 2.4 & 2.5], will also be of use. Proposition 2.11. Let K be a 1-unconditional isotropic convex body in Rn . Then p √ √ [−LK / 2, LK / 2]n ⊂ K ⊂ 3/2nB1n . In fact, we will use only the right-most inclusion. 8 Proof of Corollary 1.2. Let β > 0. Apply Lemma 2.10 with Mn := q 2(β+1) log n so that σ(Θ1 ) ≥ 1 − 2n−β . By Proposition 2.11, n hK (θ) ≤ n−1 p 3/2n kθk∞ for each θ ∈ S . Thus p σ θ ∈ S n−1 : hK (θ) ≤ 3(β + 1)n log n ≥ σ(Θ1 ) ≥ 1 − 2n−β . Let Θ be the set from Lemma 2.1. As the proof of Proposition 1.1 shows, any element of Θ satisfies the super-Gaussian estimate (4). Thus if θ ∈ Θ ∩ Θ1 , we have p vol ({x ∈ K : |hx, θi| ≥ εhK (θ)}) ≥ vol {x ∈ K : |hx, θi| ≥ ε 3(β + 1)n log n} e 2 n log n), ≥ exp(−C Cε e = 3(β + 1)) provided that (where C p C ≤ ε 3(β + 1)n log n ≤ √ n , C log n where C is the constant from Proposition 1.1. 3 On the mean-width of an isotropic convex body For a convex body K ⊂ Rn , denote its support function by hK (θ) := sup hx, θi , x∈K (θ ∈ S n−1 ). The width of K in the direction of θ is the quantity w(K, θ) = hK (θ) + hK (−θ) and the mean-width of K is Z Z w(K) = w(K, θ)dσ(θ) = 2 hK (θ)dσ(θ). S n−1 S n−1 Suppose now that vol (K) = 1. Urysohn’s inequality (see, e.g., [25, Corol√ lary 1.4]) implies that w(K) ≥ c n, with c > 0 an absolute constant. On the other hand, a combination of results due to Figiel and TomczakJaegermann [6] and Pisier [24], implies that there exists an affine image T K of K of volume one such that √ w(T K) ≤ C n log n, where C is an absolute constant. There is an important position associated to the latter fact, namely ℓ-position, see, e.g., the survey [11, §2.3], for a discussion of the corresponding circle of ideas. 9 In this section, we discuss upper bounds for the mean-width of a convex body in isotropic position (as defined in (3)). A result known to specialists is that for any isotropic convex body K ⊂ Rn , one has w(K) ≤ Cn3/4 LK , (26) where C is an absolute constant. The latter estimate follows from Dudley’s entropy estimate as in [9, Theorem 5.6] and the covering number bound from [18, Lemma 4]; a proof is in [12]. The bound (26) can also be derived easily using more recent tools, namely results of Paouris on Lq -centroid bodies in [23] (see also [7, §2 (in particular, (2.2) and Lemma 2.5)]. For the benefit of non-specialists, we mention also that sub-Gaussian estimates such as (1) in our introduction have implications for the width of K. In particular, for θ ∈ S n−1 , define Z kh·, θikψ2 := inf s > 0 : exp |hx, θi|2 /s2 dx ≤ 2 . K One can check that h·, θi satisfies (1) if and only if kh·, θikψ2 ≤ C1 kh·, θik2 , where C1 is an absolute constant. From [22, Lemma 4.2], we have √ max{hK (θ), hK (−θ)} ≤ C n kh·, θikψ2 . Thus if one could show that “most” directions are sub-Gaussian (or nearly sub-Gaussian), then one would obtain a better bound on the mean-width. As [7] and [15] show, however, it is non-trivial to establish even the existence of one θ which exhibits sub-Gaussian tail-decay. In this section, we offer another condition, related to lower bounds for caps similar to (6), under which one can bound the mean-width. 3.1 Bounding the mean-width via random polytopes Throughout this section, we assume that K is an isotropic convex body in Rn (as in (3)), X1 , . . . , XN are independent random vectors distributed uniformly in K; KN their convex hull: KN := conv {X1 , . . . , XN } ; P the associated product measure on ⊗N i=1 K. (27) √ Lemma 3.1. Let t ≥ 1 and suppose that n < N ≤ e nt/2 . Then √ p P w(KN ) ≤ C log N LK t ≥ 1 − e− nt/2 , (28) where C > 0 is an absolute constant. Proof. Let u1 , . . . , uN be points on the sphere S n−1 . Then, using (25) in a standard way, we have √ Z C1′ log N √ max |hui , θi| dσ(θ) ≤ , (29) n S n−1 i≤N where C1′ is an absolute constant. 10 v(θ, ε) hK (θ) εhK (θ) θ 0 K Figure 1: The volume v(θ, ε) of a cap. By [23, Theorem 1.1], we have √ √ P |Xi | ≤ C2′ nLK t for each i = 1, . . . , N ≥ 1 − e− nt/2 , √ where C2′ is an absolute constant. Assume now that 0 < |Xi | ≤ C2′ nLK t ′ and write Xi = Xi / |Xi |. Then Z w(KN ) ≤ 2 max |hXi , θi| dσ(θ) i≤N S n−1 √ ≤ 2C2′ nLK t ≤C p Z S n−1 log N LK t, max Xi′ , θ dσ(θ) i≤N where we used (29) for the last inequality and C = 2C1′ C2′ . Remark 3.2. See [5, Proposition 3.3] for further observations about the mean-width of the random polytope KN ; in particular, the relation to the width of Lq -centroid bodies. Next, we use an idea of Giannopoulos and Milman from [10, Lemma 5.1]. For each ε ∈ (0, 1) and θ ∈ S n−1 , let v(θ, ε) := vol ({x ∈ K : hx, θi ≥ εhK (θ)}) , as in Figure 3.1. Lemma 3.3. Let ε > 0. Then P (hKN (θ) < εhK (θ)) ≤ exp (−N v(θ, ε)) . Proof. By definition, vol ({x ∈ K : hx, θi < εhK (θ)}) = 1 − v(θ, ε), hence P max hXj , θi < εhK (θ) = (1 − v(θ, ε))N ≤ exp(−N v(θ, ε)). j≤N 11 (30) 3.2 Sufficient conditions for bounding the meanwidth In this section, we prove that one can bound the mean-width of an isotropic convex body under a certain hypothesis; namely, that in “most” directions θ, the volume of the caps v(θ, ε) (cf. (30)) is suitably large. “Most” in this case is meant with respect to the Haar measure σ on the sphere S n−1 , and is quantified by a certain constant; expressly, let C0 be the smallest constant such that for any positive integer n and any isotropic convex body K ⊂ Rn , Z S n−1 max |hx, θi| x∈K 2 dσ(θ) !1/2 ≤ C0 Z max |hx, θi| dσ(θ), S n−1 x∈K (31) By Fact 2.8, C0 is an absolute constant. It will play a role in the formulation of the proposition. Proposition 3.4. Let n be a positive integer and K an isotropic convex body in Rn . Let α ≥ 1, ε ∈ (0, 1) and p ∈ [1, 2]. Let v(θ, ε) be the volume √ of the cap defined in (30) and C0 as in (31). If 4αεp n ≥ 1 and p 1 , (32) σ {θ ∈ S n−1 : v(θ, ε) ≥ e−αε n } ≥ 1 − 16C02 then b 3/2 ε3p/2−1 nLK , w(K) ≤ Cα (33) b is an absolute constant. where C Before proving the proposition, we give several remarks to illustrate its potential utility and emphasize the important ranges for α, ε and p. Remark 3.5. The argument from [10, Lemma 5.1] shows that for every θ ∈ S n−1 and every ε ∈ (0, 1), one has v(θ, ε) ≥ c (1 − ε)n , n2 where c > 0 is an absolute constant. But (c/n2 )(1 − ε)n ≥ e−3εn provided that log(n2 /c)/n ≤ ε ≤ 1/2. Hence (32) holds with α = 3, ε = n−1/2 , and p = 1, in which case the proposition recovers the known estimate: w(K) ≤ Cn3/4 LK , with C an absolute constant. Remark 3.6. If (32) holds with α = C4′ log n, ε= 1 , n1/4 log1/2 n one would obtain the optimal bound p w(K) ≤ C n log nLK , where C is an absolute constant. 12 p = 2, (34) Remark 3.7. Corollary 1.2 shows that (32) holds with α, ε and p as in the previous remark (34) for all 1-unconditional isotropic convex bodies (with a stronger measure estimate). Note, however, that we have used Proposition 2.11 (in particular, the upper-bound on the width) to prove Corollary 1.2. Nevertheless, this shows that (32) holds with the values in (34) for a large class of convex bodies. √ Proof of Proposition 3.4. Let t = 4αεp n so that (by assumption) t ≥ 1. √ Set N = e nt/2 and suppose that X1 , . . . , XN are independent random vectors distributed uniformly in K and, as in (27), KN is their convex hull. By Lemma 3.1, we have p w(KN ) ≤ C log N LK t (35) √ with probability at least 1 − e− nt/2 . On the other hand, we can use Lemma 3.3 and an approximation argument, as in [10, Theorem 5.2], to bound the width of K by that of KN . For convenience, denote the set appearing in (32) by A(α, ε, p). A standard volume argument shows that for any η ∈ (0, 1), there exists an η-net N ⊂ A(α, ε, p), i.e., a finite set satisfying the condition ∀θ ∈ A(α, ε, p), ∃θ0 ∈ N such that |θ − θ0 | < η, with cardinality #N ≤ (3/η)n . In particular, for η = ε/4(n + 1), let us fix one such η-net N ⊂ A(α, ε, p) with cardinality #N ≤ (12(n + 1)/ε)n . (36) Claim 3.8. ε P ∃θ ∈ A(α, ε, p) : hKN (θ) < hK (θ) ≤ P (∃θ0 ∈ N : hKN (θ0 ) ≤ εhK (θ0 )) . 2 (37) Proof of Claim 3.8. Suppose that there exists θ ∈ A(α, ε, p) such that hKN (θ) < (ε/2)hK (θ). Choose θ0 ∈ N such that |θ − θ0 | < η. Note that LK B2n ⊂ K ⊂ (n + 1)LK B2n ; (38) see [13, Theorem 4.1]. The claim then follows from hKN (θ0 ) ≤ hKN (θ) + hKN (θ0 − θ) ≤ (ε/2)hK (θ) + hK (θ0 − θ) ≤ (ε/2)hK (θ0 ) + (ε/2)hK (θ − θ0 ) + hK (θ0 − θ) ≤ (ε/2)hK (θ0 ) + 2(n + 1)LK η ≤ (ε/2)hK (θ0 ) + 2(n + 1)ηhK (θ0 ) ≤ εhK (θ0 ) 13 (by (38)) (by (38)) (η = ε/(4(n + 1))). Claim 3.8 and Lemma 3.3 yield P ∃ θ ∈ A(α, ε, p) : hKN (θ) < ε hK (θ) ≤ #N max exp(−N v(θ0 , ε)) θ0 ∈N 2 n p 12(n + 1) ≤ exp −eαε n/2 . ε At this point a remark on the possible range of ε is in order. Our desired conclusion (33) is a triviality if α3/2 ε3p/2−1 > 1 (by the diameter bound (38)); hence we may assume α3/2 ε3p/2−1 ≤ 1, in which case our assump√ tion 4αεp n ≥ 1 yields the restriction ε ≥ 1/(8n3/4 ). Thus the latter probability is at most √ p (96(n + 1))2n exp −eαε n/2 ≤ exp 2n log(96(n + 1)) − e n/8 √ ≤ exp −(1/2)e n/8 , provided that n satisfies 2n log(96(n + 1)) ≤ (1/2)e √ n/8 . Therefore hK (θ) ≤ 2ε−1 hKN (θ) for each θ ∈ A(α, ε, p) √ with probability at least 1 − exp −e n/8 /2 . (39) Thus if KN satisfies both (35) and (39), we have Z Z hK (θ)dσ(θ) ≤ 2ε−1 hKN (θ)dσ(θ) A(α,ε,p) A(α,ε,p) ≤ 2ε−1 w(KN ) p ≤ 2ε−1 C log N LK t. While on the compliment A(α, ε, p)c = S n−1 \A(α, ε, p), Z A(α,ε,p)c hK (θ)dσ(θ) ≤ Z S n−1 ≤ C0 w(K) max |hx, θi| x∈K p 2 1/2 dσ(θ) p σ(A(α, ε, p)c ) σ(A(α, ε, p)c ) ≤ w(K)/4. Combining the latter estimates, Z Z w(K) = 2 hK (θ)dσ(θ) + 2 A(α,ε,p) ≤ 4ε−1 C hence p hK (θ)dσ(θ) A(α,ε,p)c log N LK t + w(K)/2, b 3/2 ε3p/2−1 nLK , w(K) ≤ Cα b an absolute constant. with C 14 Acknowledgements I thank my supervisor, Professor Nicole Tomczak-Jaegermann, for valuable feedback and continued guidance throughout my Ph.D. program. Most of the research for this article took place while I was visiting the University of Athens, from February to May 2008, as an Early-Stage Researcher with the Phenomena in High Dimensions (PHD) European network (MRTN-CT-2004-511953). I am grateful to the institution, the PHD network, and, especially, my host Professor Apostolos Giannopoulos for his gracious hospitality and help with all things mathematical and otherwise. 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Peter Pivovarov Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta Edmonton, AB, Canada T6G-2G1 ppivovarov@math.ualberta.ca 17