Ψ2 -Estimates for Linear Functionals on Zonoids G. Paouris Department of Mathematics, University of Crete, Iraklion, Greece
[email protected] Summary. Let K be a convex body in Rn with centre of mass at the origin and √ volume |K| = 1. We prove that if K ⊆ α nB2n where B2n is the Euclidean unit n−1 ball, then there exists θ ∈ S such that ·, θLψ2 (K) ≤ cα·, θL1 (K) ,
(∗)
where c > 0 is an absolute constant. In other words, “every body with small diameter has ψ2 -directions”. This criterion applies to the class of zonoids. In the opposite direction, we show that if an isotropic convex body K of volume 1 satisfies (∗) for √ every direction θ ∈ S n−1 , then K ⊆ Cα2 n log nB2n , where C > 0 is an absolute constant.
1 Introduction We shall work in Rn which is equipped with a Euclidean structure ·, ·. The Euclidean norm x, x1/2 is denoted by |·|. We write B2n for the Euclidean unit ball, S n−1 for the unit sphere, and σ for the rotationally invariant probability measure on S n−1 . Throughout this note we assume that K is a convex body in Rn with volume |K| = 1 and centre of mass at the origin. Given α ∈ [1, 2], the Orlicz norm f ψα of a bounded measurable function f : K → R is defined by α |f (x)| f ψα = inf t > 0 : dx ≤ 2 . (1.1) exp t K It is not hard to check that f ψα sup
f p : p ≥ 1 . p1/α
(1.2)
Let y = 0 in Rn . We say that K satisfies a ψα -estimate with constant bα in the direction of y if ·, yψα ≤ bα ·, y1 . (1.3) We say that K is a ψα -body with constant bα if (1.3) holds for every y = 0. It is easy to see that if K satisfies a ψα -estimate in the direction of y and if T ∈ SL(n), then T (K) satisfies a ψα -estimate (with the same constant) in the direction of T ∗ (y). It follows that T (K) is a ψα -body if K is a ψα body. By Borell’s lemma (see [MiS], Appendix III), every convex body K is a ψ1 -body with constant b1 = c, where c > 0 is an absolute constant.
V.D. Milman and G. Schechtman (Eds.): LNM 1807, pp. 211–222, 2003. c Springer-Verlag Berlin Heidelberg 2003
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Estimates of this form are related to the hyperplane problem for convex bodies. Recall that a convex body K of volume 1 with centre of mass at the origin is called isotropic if there exists a constant LK > 0 such that x, θ2 dx = L2K (1.4) K
for all θ ∈ S n−1 . Every convex body K with centre of mass at the origin has an isotropic image under GL(n) which is uniquely determined up to orthogonal transformations (for more information on the isotropic position, see [MiP]). It follows that the isotropic constant LK is an invariant for the class {T (K) : T ∈ GL(n)}. The hyperplane problem asks if every convex body of volume 1 has a hyperplane section through its centre of mass with “area” greater than an absolute constant. An affirmative answer to this question is equivalent to the following statement: there exists an absolute constant C > 0 such that LK ≤ C for every isotropic convex body K. √ Bourgain [Bou] has proved that LK ≤ c 4 n log n for every origin symmetric isotropic convex body K in Rn (the same estimate holds true for non-symmetric convex bodies as well; see [D2] and [P]). Bourgain’s argument shows that if K is a ψ2 -body with constant b2 , then LK ≤ cb2 log n where c > 0 is an absolute constant. Examples of ψ2 -bodies are given by the ball and the cube in Rn . Alesker [A] has proved that the Euclidean norm satisfies a ψ2 -estimate: there exists an absolute constant C > 0 such that |x|2 dx ≤ 2 (1.5) exp C 2 I22 K for every isotropic convex body K in Rn , where I22 = K |x|2 dx. It is not clear if every isotropic convex body satisfies a good ψ2 -estimate for most directions θ ∈ S n−1 ; for a related conjecture, see [AnBP]. On the other hand, to the best of our knowledge, even the existence of some good ψ2 -direction has not been verified in full generality. This would correspond to a sharpening of Alesker’s result. Bobkov and Nazarov [BoN] have recently proved that every 1-unconditional and isotropic convex body satisfies a ψ2 -estimate with constant c in the direction y = (1, 1, . . . , 1), where c > 0 is an absolute constant. The purpose of this note is to establish an analogous fact for zonoids. Theorem 1.1. There exists an absolute constant C > 0 with the following property: For every zonoid Z in Rn with volume |Z| = 1, there exists θ ∈ S n−1 such that 1/p √ p |x, θ| dx ≤C p |x, θ|dx Z
for every p ≥ 1.
Z
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The proof of Theorem 1.1 is presented in Section 2. The argument shows n that the same is true for every convex body √ in R which has a linear image of volume 1 with diameter of the order of n (we call these “bodies with small diameter”). In Section 3 we show that zonoids belong to this class. In the opposite direction, we show that every ψ2 -isotropic convex body has small diameter. More precisely, in Section 4 we prove the following. Theorem 1.2. Let K be an isotropic convex body in Rn . Assume that K is a ψ2 -body with constant b2 . Then, √ K ⊆ Cb22 n log nB2n , where C > 0 is an absolute constant. The letters c, c1 , c2 , c etc. denote absolute positive constants, which may change from line to line. Wherever we write a b, this means that there exist absolute constants c1 , c2 > 0 such that c1 a ≤ b ≤ c2 a. We refer the reader to the books [MiS], [Pi] and [S] for standard facts that we use in the sequel. We thank the referee for suggestions that improved the presentation and some estimates.
2 Bodies with Small Diameter We say that a convex body K in Rn with √ centre of mass at the origin has “small diameter” if |K| = 1 and K ⊆ α nB2n , where α is “well bounded”. Note that a convex body has a linear image with small diameter if and only if its polar body has bounded volume ratio. Our purpose is to show that bodies with small diameter have “good” ψ2 -directions. Our first lemma follows by a simple computation. Lemma 2.1. For every p ≥ 1 and every x ∈ Rn , 1/p √ p p |x|. |x, θ| σ(dθ) √ p+n S n−1 Proof. Observe that n p n |x, y| dy = |B2 | |x, θ|p σ(dθ). n + p S n−1 B2n On the other hand, p p |x, y| dy = |x| B2n
B2n
2|B2n−1 |
|e1 , y|p dy
1
tp (1 − t2 )(n−1)/2 dt 0p+1 n+1 Γ Γ 2 p+n+2 2 . = |B2n−1 | · |x|p Γ 2
=
p
· |x|
(2.1)
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, we get Since |B2k | = π k/2 /Γ k+2 2 p+1 1 n + p Γ 2 Γ n+2 p 2 |x|p . |x, θ| σ(dθ) = √ π n Γ p+n+2 S n−1 2 The result follows from Stirling’s formula.
Lemma 2.2. Let K be a convex body in Rn with volume |K| = 1 and centre of mass at the origin. Then, |x, θ| dx ≥ c1 ≥ 1 − 2−n , σ θ ∈ S n−1 : K
where c1 > 0 is an absolute constant. Proof. The Binet ellipsoid E of K is defined by θ2E = x, θ2 dx = MK θ, θ,
K
where MK = K xi xj dx is the matrix of inertia of K (see [MiP]). It is easily checked that det MK = det MT K for every T ∈ SL(n), and this implies that |E| = (det MK )−1/2 = L−n θ−n E σ(dθ) = K . |B2n | S n−1 Then, Markov’s inequality shows that 1 σ θ ∈ S n−1 : θE ≥ LK /2 ≥ 1 − n . 2 Since LK ≥ c and ·, θ1 ·, θ2 (see [MiP]), the result follows.
n Lemma 2.3. Let K be a convex body in R√ with volume |K| = 1 and centre of mass at the origin. Assume that K ⊆ α nB2n . Then, 2 |x, θ| exp dxσ(dθ) ≤ 2, c2 α S n−1 K
where c2 > 0 is an absolute constant. Proof. For every s > 0 we have 2 ∞ 1 |x, θ| exp dxσ(dθ) = 1 + |x, θ|2k σ(dθ)dx. s k!s2k K S n−1 S n−1 K k=1
From Lemma 2.1 we see that this is bounded by k ∞ ∞ 2k c · 2k cα 1 2k 1+ |x| dx ≤ 1 + , k!s2k 2k + n s K k=1
k=1
where c, c > 0 are absolute constants. We conclude the proof taking s = c2 α where c2 = 2c .
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An application of Markov’s inequality gives the following. Corollary 2.1. Let K be a convex body in Rn with √ volume |K| = 1 and centre of mass at the origin. Assume that K ⊆ α nB2n . Then, for every A > 2 we have
2 |x, θ| 2 n−1 : exp dx < A > 1 − , σ θ∈S c2 α A K
where c2 > 0 is the constant from Lemma 2.3.
Theorem 2.1. Let K be a convex body in √ Rn with volume |K| = 1 and centre of mass at the origin. Assume that K ⊆ α nB2n . There exists θ ∈ S n−1 such that 1/p √ p |x, θ| dx ≤ Cα p |x, θ|dx K
K
for every p > 1, where C > 0 is an absolute constant. Proof. Choose A = 4. Using the inequality ez > z k /k! (z > 0), Lemma 2.2 and Corollary 2.1 we see that with probability greater than 12 − 21n a direction θ ∈ S n−1 satisfies 2 |x, θ| |x, θ|dx ≥ c1 and exp dx < 4. c2 α K K It follows that
K
|x, θ|2k dx ≤ 4k!(c2 α)2k
for every k ≥ 1, and hence K
2k
|x, θ| dx
1 2k
√
c √ ≤ cα 2k ≤ α 2k c1
K
|x, θ|dx.
This is the statement of the theorem for p = 2k. The general case follows easily.
Remarks. (a) Bourgain’s argument in [Bou] shows that LK is bounded by a power of log n for every convex body K in Rn if the following statement holds true: If an isotropic √ convex body W in Rn is contained in the centered Euclidean ball of radius α nLW , then W is a ψ2 -body with constant O(αs ). Lemma 2.3 shows that, under the same assumptions, “half” of the directions are ψ2 -directions for W , with constant cα. (b) It can be also easily proved that convex bodies with small diameter have large hyperplane sections (this can be verified in several other ways, but the argument below gives some estimate on the distribution of the volume of their (n − 1)-dimensional sections).
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Proposition 2.1. Let K be a convex body in Rn with √ volume |K| = 1 and centre of mass at the origin. Assume that K ⊆ α nB2n . Then, for every t > 0 we have 2 c3 ≥ 1 − 2e−t , σ θ ∈ S n−1 : |K ∩ θ⊥ | ≥ tα where c3 > 0 is an absolute constant. Proof. Applying Jensen’s inequality to Lemma 2.3, we get K |x, θ| dx 2 σ(dθ) ≤ 2. exp c2 α S n−1 Markov’s inequality shows that 2 |x, θ| dx ≥ c2 αt ≤ 2e−t σ θ ∈ S n−1 : K
for every t > 0. On the other hand, it is a well-known fact (see [MiP] for the symmetric case) that if K has volume 1 and centre of mass at the origin, then 1 (2.2) |x, θ| dx |K ∩ θ⊥ | K for every θ ∈ S n−1 . This completes the proof.
3 Positions of Zonoids We first introduce some notation and recall basic facts about zonoids. The support function of a convex body K is defined by hK (y) = maxx∈K x, y for all y = 0. The mean width of K is given by hK (u)σ(du). w(K) = 2 S n−1
We say that K has minimal mean width if w(K) ≤ w(T K) for every T ∈ SL(n). Recall also the definition of the area measure σK of a convex body K: for every Borel V ⊆ S n−1 we have σK (V ) = ν x ∈ bd(K) : the outer normal to K at x is in V , where ν is the (n − 1)-dimensional surface measure on K. It is clear that σK (S n−1 ) = A(K), the surface area of K. We say that K has minimal surface area if A(K) ≤ A(T K) for every T ∈ SL(n). A zonoid is a limit of Minkowski sums of line segments in the Hausdorff metric. Equivalently, a symmetric convex body Z is a zonoid if and only if its
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polar body is the unit ball of an n-dimensional subspace of an L1 space; i.e. if there exists a positive measure µ (the supporting measure of Z) on S n−1 such that 1 ◦ xZ = |x, y|µ(dy). (3.1) 2 S n−1 The class of zonoids coincides with the class of projection bodies. Recall that the projection body ΠK of a convex body K is the symmetric convex body whose support function is defined by hΠK (θ) = |Pθ (K)|,
θ ∈ S n−1 ,
(3.2)
where Pθ (K) is the orthogonal projection of K onto θ⊥ . From the integral representation 1 |u, θ| dσK (u) (3.3) |Pθ (K)| = 2 S n−1 which is easily verified in the case of a polytope and extends to any convex body K by approximation, it follows that the projection body of K is a zonoid whose supporting measure is σK . Moreover, if we denote by Cn the class of symmetric convex bodies and by Z the class of zonoids, Aleksandrov’s uniqueness theorem shows that the Minkowski map Π : Cn → Z with K → ΠK, is injective. Note also that Z is invariant under invertible linear transformations (in fact, Π(T K) = (T −1 )∗ (ΠK) for every T ∈ SL(n)) and closed in the Hausdorff metric. For more information on zonoids, see [S] and [BouL]. We shall see that three natural positions of a zonoid have small diameter in the sense of Section 2. The proof makes use of the isotropic description of such positions which allows the use of the Brascamp-Lieb inequality. 1. Lewis position: A result of Lewis [L] (see also [B]) shows that every zonotope Z has a linear image Z1 (the “Lewis position” of Z) with the following property: there exist unit vectors u1 , . . . , um and positive real numbers c1 , . . . , cm such that m hZ1 (x) = cj |x, uj | j=1
and I=
m
cj uj ⊗ uj ,
j=1
where I denotes the identity operator in Rn . Using the Brascamp-Lieb inequality, Ball proved in [B] that, under these conditions, |Z1◦ | ≤
2n n!
and
B2n ⊆
√
nZ1◦ .
The reverse Santal´o inequality for zonoids (see [R] and [GoMR]) implies that
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|Z1 | ≥ 2n
Z1 ⊆
and
This shows that diam(Z1 ) ≤
√
√
nB2n .
n|Z1 |1/n .
(3.4) (A)
B2n
2. Lowner position: Assume that is the ellipsoid of minimal volume containing a zonoid Z2 . Let Z1 be the Lewis position of Z2 . Then, √ | nB2n | |B2n | ≤ . (3.5) |Z2 | |Z1 | Now, (3.5) and (3.4) show that diam(Z2 ) ≤ 2 ≤ |Z1 |1/n ≤
√
n|Z2 |1/n .
(B)
3. Minimal mean width position: Assume that Z3 = ΠK is a zonoid of volume 1 which has minimal mean width. The results of [GM1] and [GMR] show that the area measure σK is isotropic, i.e. A(K) (3.6) u, θ2 dσK (u) = n S n−1 for every θ ∈ S n−1 , where A(K) is the surface area of K. Moreover, a result of Petty [Pe] shows that K has minimal surface area. Now, an application of the Cauchy-Schwarz inequality and (3.6) show that 1 A(K) hZ3 (θ) = |θ, u|dσK (u) ≤ √ 2 S n−1 2 n for every θ ∈ S n−1 . We will use the following fact from [GP]: Lemma 3.1. If K has minimal surface area, then
It follows that hZ3 (θ) ≤
√
A(K) ≤ n|ΠK|1/n . n/2 for every θ ∈ S n−1 . In other words, √ diam(Z3 ) ≤ n|Z3 |1/n .
(C)
The preceding discussion shows that zonoids have positions with small diameter. More precisely, we have the following statement. Theorem 3.1. Let Z be a zonoid in Lewis or Lowner or minimal mean width position. Then, √ diam(Z) ≤ n|Z|1/n .
It follows that the results of Section 2 apply to the class of zonoids: every zonoid has ψ2 -directions in the sense of Theorem 1.1. Remark. We do not know if isotropic zonoids √ have small diameter. One can check that their mean width is bounded by c n (it is of the smallest possible order).
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4 Isotropic ψ2 -Bodies have Small Diameter The purpose of this last section is to show that a convex body is a ψ2 -body only if its isotropic position has small diameter. More precisely, we prove the following. Theorem 4.1. Let K be an isotropic convex body in Rn . Assume that K is a ψ2 -body with constant b2 . Then, √ K ⊆ Cb22 n log nB2n , where C > 0 is an absolute constant. The proof will follow from two simple lemmas. The idea for the first one comes from [GM2]. Lemma 4.1. Let K be a convex body in Rn with volume 1 and centre of mass at the origin. Then, for every θ ∈ S n−1 , Γ (p + 1)Γ (n) max hpK (θ), hpK (−θ) . |x, θ|p dx ≥ 2eΓ (p + n + 1) K Proof. Consider the function fθ (t) = |K ∩(θ⊥ +tθ)|. Brunn’s principle implies 1/(n−1) that fθ is concave. It follows that n−1 t fθ (0) fθ (t) ≥ 1 − hK (θ) for all t ∈ [0, hK (θ)]. Therefore, hK (θ) |x, θ|p dx = tp fθ (t)dt + K
0
0
hK (−θ)
tp f−θ (t)dt
n−1 t ≥ t 1− fθ (0)dt hK (θ) 0 n−1 hK (−θ) t + tp 1 − fθ (0)dt hK (−θ) 0
1 p+1 = fθ (0) hp+1 (θ) + h (−θ) sp (1 − s)n−1 ds K K
hK (θ)
p
0
Γ (p + 1)Γ (n) p+1 = (θ) + h (−θ) fθ (0) hp+1 K K Γ (p + n + 1) Γ (p + 1)Γ (n) ≥ fθ (0) hK (θ) + hK (−θ) 2Γ (p + n + 1) · max hpK (θ), hpK (−θ) . Since K has its centre of mass at the origin, we have fθ ∞ ≤ efθ (0) (see [MM]), and hence
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1 = |K| =
hK (θ)
−hK (−θ)
This completes the proof.
fθ (t)dt ≤ e hK (θ) + hK (−θ) fθ (0).
Lemma 4.2. Let K be a convex body in Rn with volume 1 and centre of mass at the origin. For every θ ∈ S n−1 , ·, θψ2 ≥
c max{hK (θ), hK (−θ)} √ , n
where c > 0 is an absolute constant. Proof. Let θ ∈ S n−1 and define Ip (θ) :=
K
1/p
|x, θ|p dx
for every p ≥ 1. Then, (1.2) shows that ·, θψ2 ≥
cIn (θ) √ . n
From Lemma 4.1 we easily see that In (θ) max{hK (θ), hK (−θ)} and the result follows.
Proof of Theorem 4.1. Since K is a ψ2 -body with constant b2 , Lemma 4.2 shows that chK (θ) √ ≤ ·, θψ2 ≤ b2 ·, θ1 n for every θ ∈ S n−1 . Since K is isotropic, we have ·, θ1 ≤ LK for every θ ∈ S n−1 . Bourgain’s argument in [Bou] (see also [D1]) together with the ψ2 -assumption show that LK ≤ c b2 log n. This implies that
√ K ⊆ Cb22 n log nB2n .
Theorem 4.1 shows that ψ2 -bodies belong to a rather restricted class (their polars have at most logarithmic volume ratio). It would be interesting to decide if zonoids are ψ2 -bodies or not.
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