A Probabilistic Approach to the Geometry of the ℓ_{p}^{n} Ball

Franck Barthe

CNRS UMR8050-Laboratoire d’Analyse et de Mathématiques Appliquées

Universités de Marne-la-Vallée et de Paris 12-Val-de-Marne Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, cedex 2, France

barthe@math.univ-mlv.fr

Olivier Guédon Equipe d’Analyse Fonctionnelle de l’Institut de Mathématiques Université Paris 6 Paris 75005, France guedon@moka.ccr.jussieu.fr

Shahar Mendelson Research School of Information Sciences and Engineering The Australian National University Canberra, ACT 0200, Australia shahar.mendelson@anu.edu.au

Assaf Naor Theory Group Microsoft Research Redmond, WA 98052, USA anaor@microsoft.com

July 1, 2003

Abstract

This article investigates, by probabilistic methods, various geometric questions on B_{p}^{n}, the unit ball of ℓ_{p}^{n}. We propose realizations in terms of independent random variables of several distributions on B_{p}^{n}, including the normalized volume measure. These representations allow us to unify and extend the known results of the sub-independence of coordinate slabs in B_{p}^{n}. As another application, we compute moments of linear functionals on B_{p}^{n} which gives sharp constants in Khinchine’s inequalities on B_{p}^{n} and determines the ψ_{2}-constant of all directions on B_{p}^{n}. We also study the extremal values of several Gaussian averages on sections of B_{p}^{n} (including mean width and ℓ-norm), and derive several monotonicity results as p varies. Applications to balancing vectors in ℓ_{2} and to covering numbers of polyhedra complete the exposition.

1 Introduction

For p > 0 and a sequence of real numbers x = (x_{i})_{i=1}^{∞} denote ∥x∥_{p} = (∑_{i=1}^{∞} |x_{i}|^{p})^{1/p}. For p = ∞ we set ∥x∥_{∞} = sup_{i∈ℕ} |x_{i}|. The space of all infinite sequences x with ∥x∥_{p} < ∞ is denoted ℓ_{p}. Similarly, the space ℝ^{n} equipped with the quasi-norm ∥ · ∥_{p} is denoted ℓ_{p}^{n}. Finally, the unit balls of ℓ_{p}^{n} and ℓ_{p} are defined as B_{p}^{n} = {x ∈ ℝ^{n}; ∥x∥_{p} ≤ 1} and B_{p} = {x ∈ ℝ^{ℕ}; ∥x∥_{p} ≤ 1}, respectively.

The geometry of ℓ_{p}^{n} spaces in general, and the geometry of the ℓ_{p}^{n} balls in particular, has been intensively investigated in the past decades. A particular topic of interest has been the evaluation of the extremal volumes of sections and projections of B_{p}^{n}. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts, some of which will be described below. The purpose of the present article is to obtain several new results of this flavor. We represent various geometric parameters of B_{p}^{n} probabilistically, and apply methods from Probability Theory to estimate them.

In Section 2 we introduce representations in terms of independent random variables of some distributions on B_{p}^{n}, including the volume measure on B_{p}^{n}. Obtaining concrete realizations of the (normalized) volume measure on a general convex body K ⊂ ℝ^{n} seems to be a hopeless task. For general bodies one is therefore reduced to hunting for approximations, and this has been successfully achieved via Markov chain methods by Kannan-Lovasz-Simonovits [16] (which paper is actually the last in a long list of articles obtaining similar approximate representations. We refer to [16] and the references therein for an accurate historic depiction of the subject). The simpler structure of B_{p}^{n} allows us to give the following representation of the volume measure (which extends to p > 0 classical results for p ∈ {1, 2, ∞}).

Theorem 1. Let g_{1}, …, g_{n} be i.i.d. random variables with density 1/(2Γ(1 + 1/p))e^{−|t|^{p}} (t ∈ ℝ), and let Z be an exponential random variable independent of g_{1}, …, g_{n} (i.e. the density of Z is e^{−t}, t ≥ 0). Denote G = (g_{1}, …, g_{n}) ∈ ℝ^{n} and consider the random vector

V = \frac{G}{(∑_{i=1}^{n} |g_{i}|^{p} + Z)^{1/p}} .

Then V generates the normalized volume measure on B_{p}^{n}, i.e. for every measurable A ⊂ ℝ^{n},

P (V ∈ A) = \frac{vol(A ∩ B_{p}^{n})}{vol(B_{p}^{n})}.

Section 2.1 provides a simple probabilistic perspective to the sub-independence of coordinate slabs on B_{p}^{n}. This remarkable fact was originally proved by Ball-Perissinaki [3] for the volume measure and in [20] for the cone measure. We establish this property for more general distributions, combining an extension of Theorem 1 with arguments similar to the proof of the classical FKG inequality [14].

In Section 2.2, Theorem 1 is applied to the study of the moments of linear functionals on B_{p}^{n} for p ≥ 1. Answering a question posed to us by A. Giannopoulos, we estimate the best constants in the Khinchine inequality on B_{p}^{n} and describe the so-called ψ_{2}-directions of B_{p}^{n}.

Section 3 is devoted to the analysis of the extremal values of several geometric parameters of sections of B_{p}^{n} for p > 0. A classical result of Meyer and Pajor [19] states that for every k-dimensional subspace E of ℝ^{n}, if p ≤ 2 then vol_{k}(E ∩ B_{p}^{n}) ≤ vol_{k}(B_{p}^{k}), and if p ≥ 2 then vol_{k}(E ∩ B_{p}^{n}) ≥ vol_{k}(B_{p}^{k}). In Section 3.1 we show that for every 0 ≤ α ≤ k, every 0 ≤ β ≤ p and every k-dimensional subspace E of ℝ^{n}, if 0 < p ≤ 2 then

∫_{S^{n−1}∩E} ∥x∥_{p}^{−α}dx ≤ ∫_{S^{k−1}} ∥x∥_{p}^{−α}dx  and  ∫_{S^{n−1}∩E} ∥x∥_{p}^{β}dx ≥ ∫_{S^{k−1}} ∥x∥_{p}^{β}dx,

and if 2 < p ≤ ∞ then

∫_{S^{n−1}∩E} ∥x∥_{p}^{−α}dx ≥ ∫_{S^{k−1}} ∥x∥_{p}^{−α}dx  and  ∫_{S^{n−1}∩E} ∥x∥_{p}^{β}dx ≤ ∫_{S^{k−1}} ∥x∥_{p}^{β}dx.

The case α = k in the above inequalities is just a restatement of the Meyer-Pajor theorem. The case β = p follows from the following stronger monotonicity result, proved in Section 3.1, that the mapping

p > 0 ↦ \frac{∫_{S^{n−1}∩E} ∥x∥_{p}^{p}dx}{∫_{S^{k−1}} ∥x∥_{p}^{p}dx}

is increasing in p.

Since Gaussian and spherical averages of homogeneous functions are proportional, these facts can be restated in terms of moments of Gaussian vectors. Note that the above quantities encompass useful classical parameters of the geometry of Banach spaces, such as mean width and ℓ-norm (see [25]).

The proofs appear in Section 3.1 and consist of finding probabilistic expressions of various expectations of Gaussian vectors on subspaces of ℝ^{n}, and then applying stochastic orderings to estimate them.

In Section 3.2 we apply the Brascamp-Lieb inequality to obtain estimates in the other direction.

Section 3.3 deals with the case of the cube B_{∞}^{n}. We derive the following distributional inequalities, valid for all k-dimensional subspaces E ⊂ ℝ^{n} and every r > 0

γ_{k}(rB_{∞}^{k}) ≤ γ_{E}(E ∩ rB_{∞}^{n}) ≤ γ_{k} (r\sqrt{\frac{n}{k}}B_{∞}^{k}) ,

where γ_{k}, γ_{E} denote the standard Gaussian measure on ℝ^{k} and E, respectively. The righthand side of the above inequality follows from the Brascamp-Lieb inequality, and the lefthand side from the following monotonicity result: for every k-dimensional subspace E ⊂ ℝ^{n}, the function

r > 0 ↦ \frac{γ_{E}(E ∩ rB_{∞}^{n})}{γ_{k}(rB_{∞}^{k})},

is non-increasing.

Sections 3.4 and 3.5 are devoted to applications of the previous results. Section 3.4 deals with the Komlós conjecture which asks whether there is a universal constant c > 0 such that for every x_{1}, …, x_{m} ∈ B_{2}^{n}, there are signs ε_{1}, …, ε_{m} ∈ {−1, 1} for which ∥∑_{i=1}^{m} ε_{i}x_{i}∥_{∞} ≤ c. This challenging problem remains unsolved, and the best upper bound on c, due to Banaszczyk [4], is c = O(\sqrt{log n}). In this section we show that our estimates, together with Banaszczyk’s theorem, yield an infinite dimensional version of this result, which implies in particular a better upper bound when m = o(n).

Proposition 1. There is an absolute constant C > 0 such that for every integer m > 0 and every x_{1}, …, x_{m} ∈ ℓ_{∞}, there are signs ε_{1}, …, ε_{m} ∈ {−1, 1} for which

∥∑_{i=1}^{m} ε_{i}x_{i}∥_{∞} ≤ C\sqrt{log d} · max_{1≤1≤m} ∥x_{i}∥_{2} ≤ C\sqrt{log m} · max_{1≤1≤m} ∥x_{i}∥_{2},

where d is the dimension of the linear span of x_{1}, …, x_{m}.

Section 3.5 answers a question posed to us by M. Talagrand, concerning the number of cubes required to cover a convex hull of a finite number of points in ℓ_{2}. Given two convex sets K, L ⊂ ℓ_{∞}, denote by N(K, L) the minimal number of translates of L required to cover K (this number may be infinite). Obtaining sharp bounds on this parameter is of fundamental importance in several problems in convex geometry (see e.g. [25]), probability (see e.g. [18]) and operator theory (see e.g. [24]). Given A ⊂ ℓ_{∞} we denote by absconv(A) the convex hull of A ∪ (−A). The main result of Section 3.5 is

Proposition 2. There exists an absolute constant C > 0 such that for every integer m, ε > 0 and 2 ≤ p ≤ ∞, for all x_{1}, …, x_{m} in the unit ball of ℓ_{2},

log N(absconv\{x_{1}, …, x_{m}\}, εB_{p}) ≤ C \frac{log m}{ε^{p/(p−1)}} .

Such a statement is already known for p = 2 by the results of Carl-Pajor [13]. From Schütt’s results [30] on the entropy of the identity operator between ℓ_{2}^{d} and ℓ_{p}^{d}, if the points x_{1}, …, x_{m} are assumed to be in an ambient ℓ_{∞}^{d} then such an inequality is valid with the term log m replaced by log max(m, d). Proposition 2 bounds the covering number of the polyhedron absconv{x_{1}, …, x_{m}} ⊂ B_{2}^{d} in terms of the number of its vertices, independently of the ambient dimension.

2 Representation of Measures on B_{p}^{n}

We begin by stating a probabilistic representation of the cone measure on ∂B_{p}^{n} which is due to Schechtman-Zinn [28] and independently to Rachev-Rüschendorf [27]. This representation has applications of probabilistic and geometric nature [21, 8, 20].

Let K be convex symmetric body in ℝ^{n}. Recall that the cone measure on ∂K, denoted μ_{K}, is defined for A ⊂ ∂K by

μ_{K}(A) = \frac{vol(ta; a ∈ A, 0 ≤ t ≤ 1)}{vol(K)}.

Thus, μ_{K}(A) is the volume of the cone with base A and cusp 0, normalized by the volume of K. Alternately, μ_{K} is the unique measure for which the following polar integration formula holds: for every f ∈ L_{1}(ℝ^{n}),

∫_{ℝ^{n}} f(x)dx = n · vol(K) ∫_{0}^{∞} r^{n−1} ∫_{∂K} f(rz)dμ_{K}(z)dr.

Schechtman-Zinn and Rachev-Rüschendorf proved the following

Theorem 2 ([28, 27]). Let g_{1}, …, g_{n} be i.i.d. random variables with density e^{−|t|^{p}}/(2Γ(1+ 1/p)), t ∈ ℝ. Consider the random vector G = (g_{1}, …, g_{n}) ∈ ℝ^{n}, and denote

Y = \frac{G}{∥G∥_{p}} = \frac{G}{(∑_{i=1}^{n} |g_{i}|^{p})^{1/p}} .

Then Y is independent of ∥G∥_{p}. Moreover, Y generates the measure μ_{B_{p}^{n}}, i.e. for every measurable A ⊂ ∂B_{p}^{n}, μ_{B_{p}^{n}}(A) = P (Y ∈ A).

We propose the following extension:

Theorem 3. Let G = (g_{1}, …, g_{n}) be a random vector as in Theorem 2. Let W be a nonnegative random variable with distribution h, and independent of G. Then the random vector

\frac{G}{(∥G∥_{p}^{p} + W )^{\frac{1}{p}}}

generates the measure h({0})μ_{B_{p}^{n}} + Ψ λ_{B_{p}^{n}}, where λ_{B_{p}^{n}} stands for Lebesgue’s measure restricted to B_{p}^{n}, and for x ∈ B_{p}^{n}, Ψ(x) = ψ(∥x∥_{p}), where for r ∈ [0, 1]

[Γ(1 + \frac{1}{p})]^{n} ψ(r) = \frac{1}{(1 − r^{p})^{\frac{n}{p} +1}} ∫_{(0,∞)} w^{n/p}e^{− \frac{r^{p}w}{1−r^{p}}} dh(w).  (1)

Proof. Note that the density of |g_{i}|^{p} is

\frac{d}{du}P (|g_{i}| ≤ u^{1/p}) = \frac{2u^{\frac{1}{p} −1}}{p} · \frac{1}{2Γ(1 + 1/p)}e^{−u} = \frac{1}{Γ(1/p)}u^{\frac{1}{p} −1}e^{−u}, u > 0.

In other words, |g_{i}|^{p} has a gamma(1/p, 1) distribution. By the additivity property of the gamma semigroup, the random variable ∥G∥_{p}^{p} = ∑_{i=1}^{n} |g_{i}|^{p} has a gamma(n/p, 1) distribution, i.e. its density is 1/Γ(n/p)u^{\frac{n}{p} −1}e^{−u} (u ≥ 0).

For any f ∈ L_{1}(ℝ^{n}), and conditioning on W,

𝔼f (\frac{G}{(∥G∥_{p}^{p} + W)^{1/p}} ) = ∫_{[0,∞)} 𝔼f (\frac{G}{(∥G∥_{p}^{p} + w)^{1/p}} ) dh(w).

Since G/∥G∥_{p} and ∥G∥_{p} are independent, then for every w > 0:

𝔼f (\frac{G}{(∥G∥_{p}^{p} + w)^{1/p}} ) = 𝔼f ((\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + w} )^{1/p} \frac{G}{∥G∥_{p}} ) =

= \frac{1}{Γ(n/p)} ∫_{0}^{∞} u^{\frac{n}{p} −1}e^{−u}𝔼f ((\frac{u}{u + w} )^{1/p} \frac{G}{∥G∥_{p}} ) du

= \frac{1}{Γ(n/p)} ∫_{0}^{1} (\frac{r^{p}w}{1 − r^{p}} )^{\frac{n}{p} −1} e^{− \frac{r^{p}w}{1−r^{p}}} · 𝔼f (r \frac{G}{∥G∥_{p}} ) \frac{pr^{p−1}w}{(1 − r^{p})^{2}} dr,

where we have made the change of variable \frac{u}{u+w} = r^{p}. Hence,

𝔼f (\frac{G}{(∥G∥_{p}^{p} + W)^{1/p}} ) − h(\{0\})𝔼f (\frac{G}{∥G∥_{p}} )

= \frac{p}{Γ(n/p)} ∫_{(0,∞)} w^{n/p} ∫_{0}^{1} \frac{r^{n−1}}{(1 − r^{p})^{\frac{n}{p} +1}} e^{− \frac{r^{p}w}{1−r^{p}}} · 𝔼f (r \frac{G}{∥G∥_{p}} ) dr dh(w)

= \frac{n}{Γ (\frac{n}{p} + 1)} ∫_{0}^{1} \frac{r^{n−1}}{(1 − r^{p})^{\frac{n}{p} +1}} (∫_{(0,∞)} w^{n/p}e^{− \frac{r^{p}w}{1−r^{p}}} dh(w)) 𝔼f (r \frac{G}{∥G∥_{p}} ) dr.

On the other hand, let ω be a probability measure on B_{p}^{n} with ℓ_{p}-radial density ϕ(∥x∥_{p}) (x ∈ B_{p}^{n}). By the polar coordinate integration formula for μ_{B_{p}^{n}}, the representation from Theorem 2 and the fact that vol(B_{p}^{n}) = \frac{[2Γ( \frac{1}{p} +1)]^{n}}{Γ( \frac{n}{p} +1)},

∫_{ℝ^{n}} f(x)dω(x) = nvol(B_{p}^{n}) ∫_{0}^{1} r^{n−1}ϕ(r)𝔼f (r \frac{G}{∥G∥_{p}} ) dr

= \frac{n [2Γ (\frac{1}{p} + 1)]^{n}}{Γ (\frac{n}{p} + 1)} ∫_{0}^{1} r^{n−1}ϕ(r)𝔼f (r \frac{G}{∥G∥_{p}} ) dr,

from which the result easily follows.

Since (1) holds true for dh(w) = e^{−w}1_{\{w>0\}} dw and ψ(r) = \frac{1_{[0,1]}(r)}{vol(B_{p}^{n})}, we have established Theorem 1. We now study more general distributions. By making the change of variable s = \frac{r^{p}}{1−r^{p}} in (1) obtain the following representation theorem. We refer to the book [32] for completely monotone functions and the Laplace transform.

Theorem 4. Let ν be a probability measure on ℝ^{n} with density ψ(∥x∥_{p})1_{[0,1]}(∥x∥_{p}). Assume that the function

s ↦ \frac{1}{(1 + s)^{\frac{n}{p} +1}} ψ ((\frac{s}{1 + s})^{1/p}) , s > 0

is completely monotone. Then there is a positive random variable W such that for every measurable A ⊂ ℝ^{n},

ν(A) = P (\frac{G}{(∥G∥_{p}^{p} + W)^{1/p}} ∈ A) ,

and the density of W is given by

\frac{[2Γ (\frac{1}{p} + 1)]^{n}}{w^{n/p}} L^{−1} [s ↦ \frac{1}{(1 + s)^{\frac{n}{p} +1}} ψ ((\frac{s}{1 + s})^{1/p})] (w), w > 0,

where L is the Laplace transform.

Next, we single out an interesting case for which the above theorem may be applied: when W is a gamma(α, 1) random variable, the density of W is h(w) = 1/Γ(α)w^{α−1}e^{−w}, and thus

[2Γ (\frac{1}{p} + 1)]^{n} ψ(r) = \frac{1}{Γ(α)(1 − r^{p})^{\frac{n}{p} +1}} ∫_{0}^{∞} w^{\frac{n}{p} +α−1}e^{− \frac{w}{1−r^{p}}} dw

= \frac{(1 − r^{p})^{\frac{n}{p} +α}}{Γ(α)(1 − r^{p})^{\frac{n}{p} +1}} ∫_{0}^{∞} w^{\frac{n}{p} +α−1}e^{−w}dw

= \frac{(1 − r^{p})^{α−1}Γ (\frac{n}{p} + α)}{Γ(α)}.

Corollary 3. Let W be a gamma(α, 1) random variable. Then the random vector _{\frac{G}{(∥G∥_{p}^{p}+W)^{1/p}}}

generates the measure on B_{p}^{n} with density

f(x) = \frac{Γ (\frac{n}{p} + α)}{Γ(α) [2Γ (\frac{1}{p} + 1)]^{n}} (1 − ∥x∥_{p}^{p})^{α−1}1_{[0,1]}(∥x∥_{p}).  (2)

Finally let us give a geometric interpretation of some of our representations. Fix two integers m, n and consider the orthogonal projection of the cone measure on ∂B_{p}^{n+m} onto the first n coordinates. By the Schechtman-Zinn Theorem, this measure is generated by the random vector

\frac{(g_{1}, …, g_{n})}{(∑_{i=1}^{n} |g_{i}|^{p} + ∑_{i=n+1}^{m+n} |g_{i}|^{p})^{1/p}} .

The random variable ∑_{i=n+1}^{m+n} |g_{i}|^{p} is independent of g_{1}, …, g_{n} and has a gamma(m/p, 1) distribution. Hence, the above discussion leads to the following extension of classical observations about B_{1}^{n} and B_{2}^{n} (for these sets the cone measure coincides with the better studied normalized surface measure).

Corollary 4. When p is an integer, the orthogonal projection of the cone measure on ∂B_{p}^{n+p} onto the first n coordinates is the (normalized) volume measure on B_{p}^{n}. More generally, for arbitrary p > 0, the orthogonal projection of the cone measure on ∂B_{p}^{n+m} onto the first n coordinates has density

f(x) = \frac{Γ (\frac{n+m}{p})}{Γ (\frac{m}{p}) [2Γ (\frac{1}{p} + 1)]^{n}} (1 − ∥x∥_{p}^{p})^{\frac{m}{p} −1}1_{[0,1]}(∥x∥_{p}).

2.1 An application: sub-independence of coordinate slabs

The sub-independence of coordinate slabs in B_{p}^{n} is helpful in the study of the Central Limit Problem [1, 21] and of various deviation inequalities [20, 7]. More precisely, this property is enjoyed by the normalized volume measure on B_{p}^{n}, as proved analytically in [3] and geometrically in [1]. It was established probabilistically in [20] for the cone measure on B_{p}^{n}. In this section we combine our representation results with an argument of [20] in order to derive sub-independence of coordinate slabs for a wider class of distributions. We require the following result:

Theorem 5 ([6]). Let X_{1}, …, X_{n} be independent symmetric random variables. Assume that X_{i} has density ψ_{i} = e^{−V_{i}}, where V_{i} is locally integrable. For X = (X_{1}, …, X_{n}), the random vector \frac{X}{∥X∥_{p}} is independent of the random variable ∥X∥_{p} if and only if there are b_{1}, …, b_{n} > −1 and a, c_{1}, …, c_{n} > 0 such that for every 1 ≤ i ≤ n, ψ_{i}(x) = c_{i}|x|^{b_{i}}e^{−a|x|^{p}}.

Remark: As a consequence of this characterization, setting for k ≤ n, X^{k} := (X_{1}, …, X_{k}) (where we write for simplicity X for X^{n}), it follows that the independence of \frac{X}{∥X∥_{p}} from

∥X∥_{p} guarantees for every k < n the independence of \frac{X^{k}}{∥X^{k}∥_{p}} from ∥X^{k}∥_{p}.

The following lemma was essentially proved in [20]. It was stated there for the cone measure on ∂B_{p}^{n}, but the proof carries through to the more general setting. We sketch the argument for the sake of completeness. Our geometric interest lead us to consider symmetric variables, but it is clear that the result concerns nonnegative variables.

Lemma 5. Let X_{1}, …, X_{n} be independent symmetric random variables. For i = 1, …, n− 1, assume that X_{i} has density ψ_{i} = exp(−V_{i}), where V_{i} is locally integrable. We write μ_{n} for the law of |X_{n}|. Denote X = (X_{1}, …, X_{n}), X^{n−1} = (X_{1}, …, X_{n−1}) and assume that \frac{X^{n−1}}{∥X^{n−1}∥_{p}} is independent of ∥X^{n−1}∥_{p}. Let f_{1}, …, f_{n} : [0, ∞) → [0, ∞) be nonnegative non-decreasing functions. Then

𝔼 [∏_{i=1}^{n} f_{i} (\frac{|X_{i}|}{∥X∥_{p}} )] ≤ ∏_{i=1}^{n} 𝔼f_{i} (\frac{|X_{i}|}{∥X∥_{p}} ) .

Proof. The proof is by induction on n. Assume that n > 1 and that the required inequality holds for n − 1. Conditioning on |X_{n}|,

𝔼 [∏_{i=1}^{n} f_{i} (\frac{|X_{i}|}{∥X∥_{p}} )] =

= ∫_{ℝ^{+}} 𝔼 \{[∏_{i=1}^{n−1} f_{i} (\frac{|X_{i}|}{(∥X^{n−1}∥_{p}^{p} + r^{p})^{1/p}} )] · f_{n} (\frac{r}{(∥X^{n−1}∥_{p}^{p} + r^{p})^{1/p}} )\} dμ_{n}(r).

Note that by Theorem 5, \frac{X^{n−2}}{∥X^{n−2}∥_{p}} and ∥X^{n−2}∥_{p} are independent, so that we may apply the inductive hypothesis. Denote by φ the density of ∥X^{n−1}∥_{p} and by the independence of \frac{X^{n−1}}{∥X^{n−1}∥_{p}} and ∥X^{n−1}∥_{p} it follows that for every r > 0

𝔼 \{[∏_{i=1}^{n−1} f_{i} (\frac{|X_{i}|}{(∥X^{n−1}∥_{p}^{p} + r^{p})^{1/p}} )] · f_{n} (\frac{r}{(∥X^{n−1}∥_{p}^{p} + r^{p})^{1/p}} )\} =

= ∫_{0}^{∞} φ(u)f_{n} (\frac{r}{(u^{p} + r^{p})^{1/p}} ) · 𝔼 [∏_{i=1}^{n−1} f_{i} (\frac{u}{(u^{p} + r^{p})^{1/p}} · \frac{|X_{i}|}{∥X^{n−1}∥_{p}} )] du

≤ ∫_{0}^{∞} φ(u)f_{n} (\frac{r}{(u^{p} + r^{p})^{1/p}} ) · ∏_{i=1}^{n−1} 𝔼f_{i} (\frac{u}{(u^{p} + r^{p})^{1/p}} · \frac{|X_{i}|}{∥X^{n−1}∥_{p}} ) du

For u > 0 let h_{u}(r) = f_{n} (\frac{r}{(u^{p}+r^{p})^{1/p}} ) and

k_{u}(r) = ∏_{i=1}^{n−1} 𝔼f_{i} (\frac{u}{(u^{p} + r^{p})^{1/p}} · \frac{|X_{i}|}{∥X^{n−1}∥_{p}} ) dr.

Thus h_{u} is non-decreasing and k_{u} is non-increasing and if X′_{n} is an independent copy of X_{n} then [h_{u}(|X_{n}|) − h_{u}(|X′_{n}|)] · [k_{u}(|X_{n}|) − k_{u}(|X′_{n}|)] ≤ 0 point-wise. Taking expectation of this inequality,

∫_{ℝ^{+}} h_{u}(r)k_{u}(r)dμ_{n}(r) ≤ (∫_{ℝ^{+}} h_{u}(r)dμ_{n}(r)) (∫_{ℝ^{+}} k_{u}(r)dμ_{n}(r)) ,

implying that

𝔼 [∏_{i=1}^{n} f_{i} (\frac{|X_{i}|}{∥X∥_{p}} )] ≤ ∫_{ℝ^{+}} ∫_{0}^{∞} φ(u)h_{u}(r)k_{u}(r) dudμ_{n}(r)

≤ ∫_{0}^{∞} φ(u) (∫_{ℝ^{+}} h_{u}(r)dμ_{n}(r)) (∫_{ℝ^{+}} k_{u}(r)dμ_{n}(r)) du
= ∏_{i=1}^{n} 𝔼f_{i} (\frac{|X_{i}|}{∥X∥_{p}} ) .

The main result of this section is contained in the following theorem.

Theorem 6. Let G = (g_{1}, …, g_{n}) be a random vector with independent coordinates with distribution e^{−|t|^{p}}/(2Γ(1 + 1/p)) dt, t ∈ ℝ. Let W be a nonnegative random variable, independent from G. Let ν be the distribution (supported on B_{p}^{n}) of the vector

\frac{G}{(∥G∥_{p}^{p} + W)^{\frac{1}{p}}} .

Then for every s_{1}, …, s_{n} > 0,

ν (⋂_{i=1}^{n}{|x_{i}| ≥ s_{i}}) ≤ ∏_{i=1}^{n} ν({|x_{i}| ≥ s_{i}}).

Proof. Assume that ε is a random variable independent of G and W which takes the values +1, −1 with probability 1/2. We set X = (g_{1}, …, g_{n}, εW^{1/p}) ∈ ℝ^{n+1}. By Theorem 2 \frac{G}{∥G∥_{p}} and ∥G∥_{p} are independent, so we can apply Lemma 5 to X, with f_{i}(x) = 1_{[s_{i},∞)}(x) for i = 1, …, n and f_{n+1} = 1. Hence,

P (⋂_{i=1}^{n} \{\frac{|g_{i}|}{(∥G∥_{p}^{p} + W)^{1/p}} ≥ s_{i}\}) ≤ ∏_{i=1}^{n} P (\{\frac{|g_{i}|}{(∥G∥_{p}^{p} + W)^{1/p}} ≥ s_{i}\}) .

Remarks:

1) By the very same proof, one can see that the conclusion of Lemma 5 holds for nonnegative, non-increasing functions. Thus Theorem 6 also holds for symmetric slabs {|x_{i}| ≤ s_{i}}.

2) We have obtained subindependence of coordinate slabs for a class of measures on B_{p}^{n}, described in Theorem 3. This unifies the previously known occurrences of such subindependence, since the cone measure μ_{p}^{n} and the normalized volume measure on B_{p}^{n} belong to this class. We obtain new concrete examples, as the measures ν_{α} with density

f_{α}(x) = \frac{Γ (\frac{n}{p} + α)}{Γ(α) [2Γ (\frac{1}{p} + 1)]^{n}} (1 − ∥x∥_{p}^{p})^{α−1}1_{[0,1]}(∥x∥_{p}).

Since these measures ν_{α} are isotropic, an immediate consequence of Theorem 6 is that they enjoy the Central Limit Property in the sense that Theorem 5 of [21] holds for them. We refer to that paper for details.

2.2 An application: moment inequalities on B_{p}^{n} for p ≥ 1

In what follows, given two sequences of positive real numbers (a_{i})_{i∈I}, (b_{i})_{i∈I} the notation a_{i} ∼ b_{i} refers to the fact that there are constants c and C such that for all i ∈ I, ca_{i} ≤ b_{i} ≤ Ca_{i}. We emphasize that such c, C are always absolute numerical constants.

We can relate moments of linear functionals on B_{p}^{n} to moments of linear functionals of the random vector G = (g_{1}, …, g_{n}) with independent coordinates with distribution e^{−|t|^{p}}/(2Γ(1 + 1/p)) dt:

Lemma 6. For every integer n ≥ 1, every p, q ≥ 1 and every a ∈ ℝ^{n} one has

(\frac{1}{vol(B_{p}^{n})} ∫_{B_{p}^{n}} |∑_{i=1}^{n} a_{i}x_{i}|^{q} dx)^{1/q} ∼ \frac{1}{(max\{n, q\})^{1/p}} (𝔼| ∑_{i=1}^{n} a_{i}g_{i}|^{q})^{1/q} .

Proof. Denote a = (a_{1}, …, a_{n}). By the probabilistic representation of the volume measure on B_{p}^{n} established in Theorem 1,

\frac{1}{vol(B_{p}^{n})} ∫_{B_{p}^{n}} |∑_{i=1}^{n} a_{i}x_{i}|^{q} dx = 𝔼 |⟨\frac{G}{(∥G∥_{p}^{p} + Z)^{1/p}} , a⟩|^{q}

= 𝔼 [(\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + Z} )^{q/p} |⟨\frac{G}{∥G∥_{p}} , a⟩|^{q}]

= [𝔼 (\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + Z} )^{q/p}] · [𝔼 |⟨\frac{G}{∥G∥_{p}} , a⟩|^{q}]

= [𝔼 (\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + Z} )^{q/p}] · \frac{𝔼 |⟨G, a⟩|^{q}}{𝔼∥G∥_{p}^{q}} ,

where we have used the independence of \frac{G}{∥G∥_{p}} and ∥G∥_{p}. Applying this identity to a = (1, 0, …, 0) yields

\frac{1}{𝔼∥G∥_{p}^{q}} [𝔼 (\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + Z} )^{q/p}] = \frac{1}{vol(B_{p}^{n})𝔼|g_{1}|^{q}} ∫_{B_{p}^{n}} |x_{1}|^{q}dx.

Now, 𝔼|g_{1}|^{q} = \frac{Γ( \frac{q+1}{p} +1)}{(q+1)Γ( \frac{1}{p} +1)}, and for every p, q ≥ 1

\frac{1}{vol(B_{p}^{n})} ∫_{B_{p}^{n}} |x_{1}|^{q}dx = \frac{2vol(B_{p}^{n−1})}{vol(B_{p}^{n})} ∫_{0}^{1} u^{q}(1 − u^{p})^{\frac{n−1}{p}} du

= \frac{2 [2Γ (\frac{1}{p} + 1)]^{n−1} Γ (\frac{n}{p} + 1)}{Γ (\frac{n−1}{p} + 1) [2Γ (\frac{1}{p} + 1)]^{n}} \frac{1}{p} ∫_{0}^{1} v^{\frac{q+1}{p} −1}(1 − v)^{\frac{n−1}{p}} dv

= \frac{Γ (\frac{n}{p} + 1)}{Γ (\frac{n−1}{p} + 1) Γ (\frac{1}{p} + 1)} · \frac{Γ (\frac{q+1}{p} + 1) Γ (\frac{n−1}{p} + 1)}{(q + 1)Γ (\frac{n+q}{p} + 1)}.

Therefore,

\frac{1}{𝔼∥G∥_{p}^{q}} [𝔼 (\frac{∥G∥_{p}^{p}}{∥G∥_{p}^{p} + Z} )^{q/p}] = \frac{Γ (\frac{n}{p} + 1)}{Γ (\frac{n+q}{p} + 1)},

and by Stirling’s formula, there are constants c, C > 0 such that for all n, q, p ≥ 1,

c\frac{1}{(max\{n, q\})^{1/p}} ≤ (\frac{Γ (\frac{n}{p} + 1)}{Γ (\frac{n+q}{p} + 1)})^{1/q} ≤ C \frac{1}{(max\{n, q\})^{1/p}} .

For independent symmetric random variable with log-concave cumulated distribution function, Kwapien and Gluskin [15] obtained an almost exact expression of moments of linear functionals. We apply their result to obtain:

Proposition 7. Let n ≥ 1 be an integer. Let p, q ≥ 1 and a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} ≥ 0. Then

(𝔼| ∑_{i=1}^{n} a_{i}g_{i}|^{q})^{1/q} ∼ q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2},

The proof of Proposition 7 requires some preparation.

Lemma 8. For every t > 0

∫_{t}^{∞} e^{−u^{p}}du ≤ \frac{e^{−t^{p}}}{pt^{p−1}} ,

and for every t ≥ 1,

∫_{t}^{∞} e^{−u^{p}}du ≥ \frac{e^{−t^{p}}}{2pt^{p−1}} .

In addition, the function t ↦ ∫_{t}^{∞} e^{−u^{p}}du is log-concave.

Proof. For every t > 0

∫_{t}^{∞} e^{−u^{p}}du ≤ ∫_{t}^{∞} \frac{u^{p−1}}{t^{p−1}} e^{−u^{p}}du = \frac{e^{−t^{p}}}{pt^{p−1}} .

To prove the reverse inequality assume that t ≥ 1. Integrating by parts,

∫_{t}^{∞} e^{−u^{p}}du = ∫_{t}^{∞} u^{1−p} · u^{p−1}e^{−u^{p}}du = \frac{e^{−t^{p}}}{pt^{p−1}} − \frac{p − 1}{p} ∫_{t}^{∞} \frac{e^{−u^{p}}}{u^{p}} du ≥ \frac{e^{−t^{p}}}{pt^{p−1}} − ∫_{t}^{∞} e^{−u^{p}}du

which implies the assertion.

Finally, set f(t) = ∫_{t}^{∞} e^{−u^{p}}du. In order to show that f is log-concave it suffices show that f′′f − (f′)^{2} ≤ 0 point-wise. Now,

f′′(t)f(t) − f′(t)^{2} = e^{−t^{p}} (pt^{p−1} ∫_{t}^{∞} e^{−u^{p}}du − e^{−t^{p}}) ≤ 0,

by the first assertion we proved.

Proof of Proposition 7. For notational convenience we write the proof for p > 1 only. The case p = 1 follows by taking limits. In what follows g denotes a random variable with density 1/(2Γ(1 + 1/p))e^{−|t|^{p}}. Let θ_{p} > 0 be such that P (θ_{p}|g| ≥ 1) = 1/e. Denote N(t) = − log P (θ_{p}|g| ≥ t) and let N^{∗}(t) be the Legendre transform of N, i.e. N^{∗}(t) = sup{ts − N(s); s > 0}. By Lemma 8, N is convex, and a result of Gluskin and Kwapien [15] states that in this case,

(𝔼 |∑_{i=1}^{n} a_{i}g_{i}|^{q})^{1/q} ∼ θ_{p} [inf \{t > 0; ∑_{i≤q} N^{∗}(\frac{qa_{i}}{t} ) ≤ q\} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2}]

where a_{1} ≥ a_{2} ≥ … ≥ a_{n} ≥ 0. We shall prove below that there exists universal constants c, c′, C, C′ > 0 such that for all p ≥ 1, c′ ≤ θ_{p} ≤ C′, for all t > 0, (N^{∗}(t))^{(p−1)/p} ≤ Ct and for all t ≥ 2, (N^{∗}(t))^{(p−1)/p} ≥ ct. Let

t_{0} = inf \{t > 0; ∑_{i≤q} N^{∗}(\frac{qa_{i}}{t} ) ≤ q\}

then it is easy to see that there exists a universal constant C > 0 such that

t_{0} ≤ Cq^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p}.

Moreover, if i_{0} is the biggest integer less than q such that qa_{i_{0}−1}/t_{0} ≥ 2 then

∑_{i≤q} N^{∗}(\frac{qa_{i}}{t_{0}} ) ≥ ∑_{i≤i_{0}−1} (\frac{Cqa_{i}}{t_{0}} )^{p/(p−1)}

which shows that

t_{0} ≥ Cq^{1/p}( ∑_{i≤i_{0}−1} a_{i}^{p/(p−1)})^{(p−1)/p} and t_{0} > \frac{qa_{i_{0}}}{2} .

It is now clear that

\begin{aligned}q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} & ≤ & q^{1/p}( ∑_{i≤i_{0}−1} a_{i}^{p/(p−1)})^{(p−1)/p} + q^{1/p}(q − i_{0} + 1)^{(p−1)/p}a_{i_{0}} \\ & ≤ & \frac{t_{0}}{C} + (\frac{q − i_{0} + 1}{q})^{(p−1)/p}t_{0} ≤ ct_{0}. \end{aligned}

To prove the bounds on θ_{p}, note that since |g| has uniformly bounded density in p, there is an absolute constant c > 0 such that for every s > 0, P (|g| ≥ s) ≥ 1−cs. If s = c^{−1}(1−e^{−1}) then P (|g| ≥ s) ≥ P (|g| ≥ 1/θ_{p}), which shows that θ_{p} ≤ s^{−1} ≤ C. On the other hand, Lemma 8 implies that there is an absolute constant c′ for which P (|g| ≥ c′) ≤ 1/e = P (|g| ≥ 1/θ_{p}), and thus θ_{p} ≥ 1/c′.

It remains to prove the above mentioned bounds on N^{∗}. Lemma 8 shows that for every s ≥ θ_{p},

N(s) ≥ \frac{s^{p}}{θ_{p}^{p}} + log p + (p − 1) log (\frac{s}{θ_{p}} ) + log Γ(1 + 1/p) ≥ (\frac{s}{θ_{p}} )^{p} .

Hence, for every s ≥ θ_{p}

ts − N(s) ≤ ts − (\frac{s}{θ_{p}} )^{p} ≤ max_{s>0} \{ts − (\frac{s}{θ_{p}} )^{p}\} ≤ \frac{θ_{p}t^{p/(p−1)}}{p^{1/(p−1)}} ≤ Ct^{p/(p−1)}.

For 0 < s < θ_{p}, st − N(s) ≤ θ_{p}t = O (t^{p/(p−1)}), and the upper bound for N^{∗} follows. The lower bound in Lemma 8 shows that there are absolute constants c, C > 1 such that

for s ≥ cθ_{p}, N(s) ≤ (\frac{Cs}{θ_{p}})^{p}. If t ≥ c^{p−1}C^{p}p/θ_{p} then for s = (\frac{θ_{p}}{C} )^{p/(p−1)} (\frac{t}{p})^{1/(p−1)} ≥ cθ_{p},

N^{∗}(t) ≥ st − N(s) ≥ st − (\frac{Cs}{θ_{p}})^{p} = (1 − \frac{1}{p}) (\frac{tθ_{p}}{C} )^{p/(p−1)} \frac{1}{p^{1/(p−1)}} .

We may therefore assume that t ≤ (cC)^{p−1}p. By our choice of θ_{p}, N(1) = 1, which implies that for all t ≥ 2,

N^{∗}(t) ≥ t − N(1) ≥ \frac{t}{2} ≥ \frac{t^{p/(p−1)}}{2cCp^{1/(p−1)}} ,

completing the proof.

The results of this sections may be combined to obtain the following exact expression, up to universal constants: for a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} ≥ 0

(\frac{1}{vol(B_{p}^{n})} ∫_{B_{p}^{n}} |∑_{i=1}^{n} a_{i}x_{i}|^{q} dx)^{1/q} ∼ \frac{q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2}}{(max\{n, q\})^{1/p}} . (3)

which virtually allows one to solve any question related to moment estimates on B_{p}^{n}.

2.2.1 Khinchine inequalities

A well known variant of Khinchine’s inequality states that for every 1 ≤ p, q < ∞ and every integer n there are A(p, q, n), B(p, q, n) > 0 such that for every (a_{1}, …, a_{n}) ∈ ℝ^{n}:

A(p, q, n) (∑_{i=1}^{n} a_{i}^{2})^{1/2} ≤ (\frac{1}{vol(B_{p}^{n})} ∫_{B_{p}^{n}} |∑_{i=1}^{n} a_{i}x_{i}|^{q} dx)^{1/q} ≤ B(p, q, n) (∑_{i=1}^{n} a_{i}^{2})^{1/2} ,

and we assume that A(p, q, n), B(p, q, n) are the best constants for which the above inequality holds for all (a_{1}, …, a_{n}) ∈ ℝ^{n}. We determine A(p, q, n) and B(p, q, n), up to absolute multiplicative constants.

Theorem 7. For every integer n and for every 1 ≤ q < ∞ and 1 ≤ p ≤ 2,

A(p, q, n) ∼ \frac{\sqrt{q}}{n^{1/p}} min \{1, \sqrt{\frac{n}{q}}\} and B(p, q, n) ∼ min \{1, (\frac{q}{n})^{1/p}\}

while for 2 < p < ∞,

A(p, q, n) ∼ min \{1, (\frac{q}{n})^{1/p}\} and B(p, q, n) ∼ \frac{\sqrt{q}}{n^{1/p}} min \{1, \sqrt{\frac{n}{q}}\} .

This is a consequence of (3) and of the following:

Lemma 9. For every a = (a_{1}, …, a_{n}) ∈ S^{n−1}, if 1 < p ≤ 2 then

\sqrt{q} max \{1, (\frac{q}{n})^{\frac{1}{p} − \frac{1}{2}} \} ≤ q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} ≤ \sqrt{2} · q^{1/p}.

If 2 < p < ∞ then

q^{1/p} ≤ q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} ≤ \sqrt{2q} min \{1, (\frac{n}{q})^{\frac{1}{2} − \frac{1}{p}} \} .

Furthermore, these inequalities are optimal, up to universal constants.

Proof. Assume that 1 < p ≤ 2. Since \sqrt{a} + \sqrt{b} ≤ \sqrt{2}\sqrt{a + b},

q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} ≤ q^{1/p}( ∑_{i≤q} a_{i}^{2})^{1/2} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} ≤ \sqrt{2} · q^{1/p}

Similarly, if q > n then

q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} = q^{1/p}( ∑_{i≤n} a_{i}^{p/(p−1)})^{(p−1)/p} ≥ \frac{q^{1/p}}{n^{\frac{1}{p} − \frac{1}{2}}}

and if q ≤ n,

q^{1/p}( ∑_{i≤q} a_{i}^{p/(p−1)})^{(p−1)/p} + \sqrt{q}( ∑_{i>q} a_{i}^{2})^{1/2} ≥ \sqrt{q}.

The fact that these inequalities are best possible up to universal constants follows by considering in each case the vectors (1, 0, …, 0), (1/\sqrt{n}, …, 1/\sqrt{n}) or (1/\sqrt{q}, …, 1/\sqrt{q}, 0, …, 0) when q ≤ n. The proof of the case p ≥ 2 is equally simple.

2.2.2 ψ_{2}-directions

We start with a few definitions. Let α ∈ [1, 2] and set μ to be a probability measure on ℝ^{n}. For a measurable function f : ℝ^{n} → ℝ, define the following Orlicz norm associated with α and μ by

∥f∥_{ψ_{α}(μ)} := inf {λ > 0; ∫ e^{|f/λ|^{α}}dμ ≤ 2}.

It is well known that ∥f∥_{ψ_{α}(μ)} ∼ sup_{q≥1} q^{− \frac{1}{α}} ( ∫ |f|^{q} dμ)^{\frac{1}{q}} . Given a vector θ in the unit sphere S^{n−1} of ℝ^{n}, one says that θ defines a ψ_{α} direction for μ with a constant C > 0 if the function f_{θ}(x) = ⟨x, θ⟩ satisfies

∥f_{θ}∥_{ψ_{α}(μ)} ≤ C( ∫ |f_{θ}|^{2} dμ)^{\frac{1}{2}} .

In other words the moment of f_{θ} of order q is bounded from above by a constant times Cq^{1/α} times the second moment of f_{θ}.

From now on consider a convex body K ⊂ ℝ^{n}, with the center of mass at the origin. Such a body is said to be a ψ_{α} body with constant C if all directions θ ∈ S^{n−1} are ψ_{α} with a constant C, with respect to the uniform probability measure on K. It follows from the Brunn-Minkowski inequality that convex bodies are ψ_{1} with a uniform constant, and any improvement on this estimate would be very useful. Note that the notion of ψ_{2}-bodies is crucial in Bourgain’s bound on the isotropy constant [12] of convex bodies. This motivated recent works on the ψ_{2}-directions of convex bodies. In fact, it is not even clear that there exists a universal constant C such that any convex body (of any dimension) admits at least one ψ_{2}-direction with constant C. This question of V. Milman was solved in very special cases such as zonoids (Paouris [23]) and unconditional bodies (Bobkov and Nazarov [11] show that the main diagonal is ψ_{2}). Thanks to equation (3) we are able to study these questions for B_{p}^{n}.

Proposition 10. There exists C > 0 such that 1. For every n ≥ 1 and every p ≥ 2, B_{p}^{n} is a ψ_{2}-body with constant C.

2. For every n ≥ 1 and every p ∈ [1, 2], B_{p}^{n} is a ψ_{p}-body with constant C.

The first point was actually a consequence of results in [7], where subindependence was also used.

Proof. Without loss of generality we consider a direction θ ∈ S^{n−1} with θ_{1} ≥ θ_{2} ≥ ⋯ ≥ θ_{n} ≥ 0. Fix q ≥ 1. Equation (3) gives, with obvious notation

\begin{matrix}(𝔼_{B_{p}^{n}}|⟨X, θ⟩|^{q}) & \frac{1}{q} \\ (𝔼_{B_{p}^{n}}|⟨X, θ⟩|^{2}) & \frac{1}{2} \end{matrix} ∼ (\frac{n}{max\{n, q\}})^{\frac{1}{p}} · (q^{\frac{1}{p}} ( ∑_{i≤q} θ_{i}^{p′})^{\frac{1}{p′}} + \sqrt{q}( ∑_{i>q} θ_{i}^{2})^{\frac{1}{2}} ) ,  (4)

where p′ = p/(p − 1).

The result now follows from obvious estimates. Indeed, since n/ max{n, q} ≤ 1, for p ≥ 2, Hölder’s inequality implies that ∥(θ_{i})_{i≤q}∥_{p′} ≤ min{n, q}^{1/2−1/p}∥(θ_{i})_{i≤q}∥_{2} ≤ q^{1/2−1/p}. Hence, the right-hand side in (4) is less than 2\sqrt{q}. For p ∈ [1, 2], it is evident that ∥(θ_{i})_{i≤q}∥_{p′} ≤ ∥(θ_{i})_{i≤q}∥_{2} ≤ 1 and thus the ratio of moments is bounded by a constant times q^{1/p}.

Next, we describe the ψ_{2} constant on B_{p}^{n} of every direction for 1 ≤ p ≤ 2.

Proposition 11. Let p ∈ [1, 2]. For any integer n ≥ 1 and θ ∈ S^{n−1}, θ is a ψ_{2}-direction of B_{p}^{n} and the best constant for which it is ψ_{2} is, up to an absolute multiplicative constants, n^{1/p−1/2}∥θ∥_{p′}.

Observe that from the above result, the direction of the main diagonal is ψ_{2}. For p = 1 we recover a result of Bobkov and Nazarov [10] (let us note that in that paper, the authors give another moment estimate for B_{1}^{n}, which can be recovered by our method, and which implies that most directions are ψ_{2−ϵ}. Moreover Bobkov and Nazarov show that these moment upper estimates for B_{1}^{n} can be transferred to isotropic unconditional convex bodies).

Proof. Assume, as we may, that θ_{1} ≥ θ_{2} ≥ ⋯ ≥ θ_{n} ≥ 0. For q < n the right-hand side of (4) is equal to

q^{\frac{1}{p}} ∥(θ_{i})_{i≤q}∥_{p′} + \sqrt{q}∥(θ_{i})_{i>q}∥_{2} ≤ \sqrt{q} (n^{\frac{1}{p} − \frac{1}{2}} ∥(θ_{i})_{i≤n}∥_{p′} + 1) ≤ 2\sqrt{q} n^{\frac{1}{p} − \frac{1}{2}} ∥(θ_{i})_{i≤n}∥_{p′},

where we used Hölder’s inequality in the form 1 = ∥(θ_{i})_{i≤n}∥_{2} ≤ n^{\frac{1}{p} − \frac{1}{2}} ∥(θ_{i})_{i≤n}∥_{p′}.

If q ≥ n the right-hand side of (4) is n^{\frac{1}{p}} ∥θ∥_{p′} ≤ \sqrt{q} n^{\frac{1}{p} − \frac{1}{2}} ∥(θ_{i})_{i≤n}∥_{p′}. For q = n, it is easy to see that the estimate cannot be improved by more than a universal factor.

3 Extremal Geometric Parameters of Sections of B_{p}^{n}, p > 0

In what follows we will denote by G a standard Gaussian vector. If E ⊂ ℝ^{n} is a kdimensional subspace then G will still stand for a standard Gaussian vector on E (which is well defined due to rotational invariance).

3.1 Bounds via stochastic ordering

In this section, we present monotonicity properties for sections of B_{p}^{n} as p > 0 varies. We follow the approach of Meyer and Pajor [19]. They proved that for a fixed vector subspace of dimension k in ℝ^{n}, the ratio Vol_{k}(E ∩B_{p}^{n})/Vol_{k}(B_{p}^{k}) is non-decreasing in p ≥ 1. This was later extended to p > 0 and to ℓ_{p}-sums of arbitrary spaces of finite dimension (see [5] and the reference therein). We are interested in Gaussian averages of the ℓ_{p}-norm on sections. Our results will recover in several ways the latter result on the volume.

We will use the notion of peaked ordering on measures. Given two absolutely continuous measures μ and ν on ℝ^{d}, one says that ν is more peaked than μ and writes μ ≺ ν if for every symmetric bounded convex set C

μ(C) ≤ ν(C).

In the following statement, we put together the properties that we need. They follow from more general results by Kanter [17].

Proposition 12. Let μ, ν be probability measures on ℝ, with even densities which are non-increasing on [0, ∞). If μ ≺ ν then for every n ≥ 1 one has μ^{⊗n} ≺ ν^{⊗n}.

The aim of the next two lemmas is to relate Gaussian averages of the ℓ_{p}-norm on subspaces to the values of some product measures. Let E ⊂ ℝ^{n} be a subspace with dim(E) = k. We denote by P_{E} the orthogonal projection from ℝ^{n} onto E and let u_{k+1}, …, u_{n} be an orthonormal basis of E^{⊥}. Set

B_{∞}(E^{⊥}) = \{x ∈ E^{⊥}; sup_{i=k+1,...,n} |⟨x, u_{i}⟩| ≤ \frac{1}{2}\}

and for ϵ > 0,

E(ϵ) = {x ∈ ℝ^{n}; x − P_{E}(x) ∈ ϵB_{∞}(E^{⊥})} .

Lemma 13. Let E be a k-dimensional subspace of ℝ^{n} and set h to be a continuous function in L_{1}(ℝ^{n}, γ_{n}), with the following property: there exist K, η > 0 such that for every x ∈ ℝ^{n} one has |h(x)| ≤ Ke^{∥x∥_{2}^{2}/(2+η)}. Then

∫_{E} h(x)dγ_{E}(x) = lim_{ϵ→0} (\frac{2π}{ϵ^{2}} )^{\frac{n−k}{2}} ∫_{E(ϵ)} h(x)dγ_{n}(x).

Proof. Fix some ϵ > 0. By the rotational invariance of the Gaussian measure,

(2π)^{n/2} ∫_{E(ϵ)} h(x)dγ_{n}(x) = ∫_{ℝ^{n}} e^{− \frac{∥x∥^{2}_{2}}{2}} 1_{E(ϵ)}(x)h(x)dm_{n}(x)
= ∫_{E×ϵB_{∞}(E^{⊥})} e^{− \frac{∥a∥^{2}_{2}}{2} − \frac{∥b∥^{2}_{2}}{2}} h(a + b)dm_{E}(a)dm_{E^{⊥}}(b)
= ϵ^{n−k} ∫_{E×B_{∞}(E^{⊥})} e^{− \frac{∥a∥^{2}_{2}}{2}} e^{−ϵ^{2} \frac{∥c∥^{2}_{2}}{2}} h(a + ϵc)dm_{k}(a)dm_{n−k}(c).

By continuity and dominated convergence, the latter integral converges when ϵ goes to zero to

vol_{n−k}(B_{∞}(E^{⊥})) ∫_{E} e^{− \frac{∥a∥^{2}_{2}}{2}} h(a)dm_{k}(a),

which gives the claimed result.

Fix 0 < p, λ < ∞, let α(p, λ) = 2 ∫_{0}^{∞} e^{−λt^{p}−t^{2}}dt and set μ_{p,λ} to be the probability measure on ℝ defined by

dμ_{p,λ}(t) = e^{−λα(p,λ)^{p}|t|^{p}−α(p,λ)^{2}t^{2}} dt.

Lemma 14. Let E be a k-dimensional subspace of ℝ^{n} and 0 < p, λ < ∞. Then

\frac{𝔼e^{− \frac{λ}{2^{p/2}} ∥G∥_{E∩B^{n}_{p}}^{p}}}{𝔼e^{− \frac{λ}{2^{p/2}} ∥G∥_{B^{k}_{p}}^{p}}} = lim_{ϵ→0} ϵ^{k−n}μ_{p,λ}^{⊗n}(E(ϵ)).

Proof. By Lemma 13,

𝔼e^{− \frac{λ}{2^{p/2}} ∥G∥_{E∩B^{n}_{p}}^{p}} = lim_{ϵ→0} (\frac{2π}{ϵ^{2}} )^{\frac{n−k}{2}} ∫_{E(ϵ)} e^{− \frac{λ∥x∥^{p}_{p}}{2^{p/2}} − \frac{∥x∥^{2}_{2}}{2}} dm_{n}(x)

= lim_{ϵ→0} (\frac{2π}{ϵ^{2}} )^{\frac{n−k}{2}} 2^{n/2}α(p, λ)^{n} ∫_{E( \frac{ϵ}{\sqrt{2}α(p,λ)} )} e^{−λα(p,λ)^{p}∥x∥_{p}^{p}−α(p,λ)^{2}∥x∥_{2}^{2}}dm_{n}(x)

= lim_{ϵ→0} 2^{k/2}α(p, λ)^{k} (\frac{2π}{ϵ^{2}} )^{\frac{n−k}{2}} μ_{p,λ}^{⊗n}(E(ϵ)).

Thus, applied to E = ℝ^{k} × {0_{n−k}} with u_{i} = e_{i} for i > k, this identity yields

𝔼e^{− \frac{λ}{2^{p/2}} ∥G∥_{B^{k}_{p}}^{p}} = lim_{ϵ→0} 2^{k/2}α(p, λ)^{k} (\frac{2π}{ϵ^{2}} )^{\frac{n−k}{2}} (∫_{−ϵ/2}^{ϵ/2} e^{−λα(p,λ)^{p}|x|^{p}−α(p,λ)^{2}x^{2}}dx)^{n−k}
= 2^{k/2}α(p, λ)^{k}(2π)^{\frac{n−k}{2}} ,

from which the required result follows.

In the forthcoming lemmas and propositions, we look for comparison results in the sense of the peaked ordering, between measures of the form μ_{p,λ}. We start with useful facts about the constants α(p, λ) which appear in the definition of μ_{p,λ}.

Lemma 15. Let λ > 0 and 0 < p < q < ∞. Then α (p, _{\frac{λ}{Γ(\frac{p+1}{2})}}) < α (q, _{\frac{λ}{Γ(\frac{q+1}{2})}}) .

Proof. By its definition,

α(p, λ) = 2 ∫_{0}^{∞} e^{−λt^{p}−t^{2}}dt = \sqrt{π} · 𝔼 exp (−\frac{λ|g|^{p}}{2^{p/2}} ) ,

where g is a standard Gaussian random variable. Recall that

𝔼|g|^{p} = \frac{2^{p/2}}{\sqrt{π}} Γ (\frac{p + 1}{2}) ,

and thus

α (p, \frac{λ}{Γ (\frac{p+1}{2})}) = \sqrt{π} · 𝔼 exp (−\frac{λ|g|^{p}}{\sqrt{π} · 𝔼|g|^{p}} ) .

Therefore, Lemma 15 follows from the following result:

Lemma 16. Fix 0 < p < q < ∞ and let X be a non-negative random variable with 𝔼X^{q} < ∞. Then for every convex function f : [0, ∞] → [0, ∞),

𝔼f (\frac{X^{p}}{𝔼X^{p}} ) ≤ 𝔼f (\frac{X^{q}}{𝔼X^{q}} ) .

Proof. Let t_{0} be defined by t_{0}^{1/p}(𝔼X^{p})^{1/p} = t_{0}^{1/q}(𝔼X^{q})^{1/q}. Clearly,

𝔼 [f (\frac{X^{p}}{𝔼X^{p}} ) − f (\frac{X^{q}}{𝔼X^{q}} )] = ∫_{0}^{t_{0}} f′(t)h(t)dt + ∫_{t_{0}}^{∞} f′(t)h(t)dt,  (5)

where h(t) = P (X^{p} ≥ t𝔼X^{p}) − P (X^{q} ≥ t𝔼X^{q}). Since h ≥ 0 on [0, t_{0}] and h ≤ 0 on [t_{0}, ∞) and ∫_{0}^{∞} h(t)dt = 0, then

∫_{0}^{t_{0}} f′(t)h(t)dt + ∫_{t_{0}}^{∞} f′(t)h(t)dt = ∫_{0}^{t_{0}}[f′(t) − f′(t_{0})]h(t)dt + ∫_{t_{0}}^{∞}[f′(t) − f′(t_{0})]h(t)dt ≤ 0,

where we have used the fact that f′ is non-decreasing. Combined with (5), this completes the proof.

Proposition 17. Let 0 < p < q, and λ_{1}, λ_{2} > 0. Then:

a) If q ≥ 2 and α(p, λ_{1}) > α(q, λ_{2}) then μ_{p,λ_{1}} ≺ μ_{q,λ_{2}}.

b) If q < 2 and α(p, λ_{1}) < α(q, λ_{2}) then μ_{p,λ_{1}} ≺ μ_{q,λ_{2}}.

c) If p < 2 and q ≥ 2 then without any restriction on λ_{1} and λ_{2}, μ_{p,λ_{1}} ≺ μ_{q,λ_{2}}.

d) If 0 < p < 2 and λ_{1} < λ_{2} then μ_{p,λ_{2}} ≺ μ_{p,λ_{1}}.

e) If p > 2 and λ_{1} < λ_{2} then μ_{p,λ_{1}} ≺ μ_{p,λ_{2}}.

Proof. Define h : [0, ∞) → ℝ by

h(a) = ∫_{0}^{a} [e^{−λ_{1}α(p,λ_{1})^{p}t^{p}−α(p,λ_{1})^{2}t^{2}} − e^{−λ_{2}α(q,λ_{2})^{q}t^{q}−α(q,λ_{2})^{2}t^{2}}] dt.

In order to prove that μ_{p,λ_{1}} ≺ μ_{q,λ_{2}} one has to show that h(a) ≤ 0 for all a ≥ 0. Note that h(0) = lim_{x→∞} h(x) = 0, and if

ψ(t) = −λ_{1}α(p, λ_{1})^{p}t^{p−2} − α(p, λ_{1})^{2} + λ_{2}α(q, λ_{2})^{q}t^{q−2} + α(q, λ_{2})^{2}

then sign(h′) = sign(ψ).

In case a), lim_{a→0} ψ(a) < 0 and lim_{a→∞} ψ(a) > 0. Hence h′ < 0 in a neighborhood of 0 and h′(a) > 0 for a large enough. If there were some a_{0} > 0 such that h(a_{0}) > 0 then it would follow that h′ must have at least 3 zeros. Thus ψ would also have 3 zeros, implying that ψ′ has at least 2 zeros. This is impossible since

ψ′(t) = −λ_{1}(p − 2)α(p, λ_{1})^{p}t^{p−3} + λ_{2}(q − 2)α(q, λ_{2})^{q}t^{q−3}

clearly has at most one zero.

Cases b) and c) are just as simple. To prove case d) one must show that the function

ψ(t) = (λ_{2}α(p, λ_{2})^{p} − λ_{1}α(p, λ_{1})^{p})t^{p−2} + α(p, λ_{2})^{2} − α(p, λ_{1})^{2}

is first positive and then negative. Since it changes signs only once, it is enough to check this at zero and infinity. Observe that

α(p, λ) = 2 ∫_{0}^{∞} e^{−λt^{p}−t^{2}}dt  and  λ^{1/p}α(p, λ) = 2 ∫_{0}^{∞} e^{−t^{p}− \frac{t^{2}}{λ^{2/p}}} dt,

so that α(p, λ) is decreasing in λ and λα(p, λ)^{p} is increasing in λ. Since p < 2 then lim_{x→0} ψ(x) = +∞ and lim_{x→∞} ψ(x) < 0. The proof of the last case is almost identical.

Proposition 18. Let E be a k-dimensional subspace of ℝ^{n} and set λ > 0. For p > 0, let

F (p) = \frac{𝔼 exp [−\frac{λ∥G∥_{E∩B^{n}_{p}}^{p}}{2^{p/2}Γ(\frac{p+1}{2})}]}{𝔼 exp [−\frac{λ∥G∥_{B^{k}_{p}}^{p}}{2^{p/2}Γ(\frac{p+1}{2})}]}.

Then F is non-decreasing on (0, 2]. Moreover for p ≥ 2 one has F (p) ≥ F (2) = 1.

Proof. Let r < 2, fix some λ > 0, let p > r, and define

λ_{1} = \frac{λ}{Γ (\frac{r+1}{2})} and λ_{2} = \frac{λ}{Γ (\frac{p+1}{2})}.

By Lemma 15 and cases b and c of Proposition 17, μ_{r,λ_{1}} ≺ μ_{p,λ_{2}}. Tensorizing and applying Proposition 12, it follows that μ_{r,λ_{1}}^{⊗n} ≺ μ_{p,λ_{2}}^{⊗n}. In particular, for every ϵ > 0,

μ_{r,λ_{1}}^{⊗n}(E(ϵ)) ≤ μ_{p,λ_{2}}^{⊗n}(E(ϵ)).

By Lemma 14

\frac{𝔼 exp [−\frac{λ∥G∥_{E∩B^{n}_{r}}^{r}}{2^{r/2}Γ(\frac{r+1}{2})}]}{𝔼 exp [−\frac{λ∥G∥_{B^{k}_{r}}^{r}}{2^{r/2}Γ(\frac{r+1}{2})}]} ≤ \frac{𝔼 exp [−\frac{λ∥G∥_{E∩B^{n}_{p}}^{p}}{2^{p/2}Γ(\frac{p+1}{2})}]}{𝔼 exp [−\frac{λ∥G∥_{B^{k}_{p}}^{p}}{2^{p/2}Γ(\frac{p+1}{2})}]},  (6)

hence F (r) ≤ F (p) holds when r < 2 and r < p.

Theorem 8. Let E be a k-dimensional subspace of ℝ^{n}. Then the function

p ↦ \frac{𝔼∥G∥_{E∩B_{p}^{n}}^{p}}{𝔼∥G∥_{B_{p}^{k}}^{p}}

is non-increasing in p > 0.

Proof. Assume that p < q ≤ 2. Both sides of (6) equal 1 for λ = 0, so the same inequality must hold between the derivatives at 0 of both sides, that is

−\frac{𝔼∥G∥_{E∩B_{p}^{n}}^{p}}{2^{p/2}Γ (\frac{p+1}{2})} + \frac{𝔼∥G∥_{B_{p}^{k}}^{p}}{2^{p/2}Γ (\frac{p+1}{2})} ≤ −\frac{𝔼∥G∥_{E∩B_{q}^{n}}^{p}}{2^{q/2}Γ (\frac{q+1}{2})} + \frac{𝔼∥G∥_{B_{q}^{k}}^{q}}{2^{q/2}Γ (\frac{q+1}{2})}.

Note that

𝔼∥G∥_{B_{p}^{k}}^{p} = 𝔼 ∑_{i=1}^{k} |g_{i}|^{p} = \frac{2k}{\sqrt{2π}} ∫_{0}^{∞} x^{p}e^{− \frac{x^{2}}{2}} dx = \frac{2^{\frac{p}{2} +1}}{\sqrt{π}} kΓ (\frac{p + 1}{2}) .

Hence, the above inequality translates to

−\frac{2k}{\sqrt{π}} · \frac{𝔼∥G∥_{E∩B_{p}^{n}}^{p}}{𝔼∥G∥_{B_{p}^{k}}^{p}} + \frac{2k}{\sqrt{π}} ≤ −\frac{2k}{\sqrt{π}} · \frac{𝔼∥G∥_{E∩B_{q}^{n}}^{q}}{𝔼∥G∥_{B_{q}^{k}}^{q}} + \frac{2k}{\sqrt{π}} ,

so that

\frac{𝔼∥G∥_{E∩B_{p}^{n}}^{p}}{𝔼∥G∥_{B_{p}^{k}}^{p}} ≥ \frac{𝔼∥G∥_{E∩B_{q}^{n}}^{q}}{𝔼∥G∥_{B_{q}^{k}}^{q}} .

It remains to deal with the case 2 ≤ p < q, which is slightly more complicated because the last proposition does not give much in this case for a fixed value of the parameter λ. However, something remains true when λ tends to zero, and thus one can pass to the limit.

Indeed, fix two numbers c_{p}, c_{q} > 0 such that

c_{p} < \frac{1}{Γ (\frac{p+1}{2})} and c_{q} > \frac{1}{Γ (\frac{q+1}{2})},

and for every λ > 0 define

f(λ) = α(p, c_{p}λ) − α(q, c_{q}λ) = 2 ∫_{0}^{∞} e^{−c_{p}λt^{p}−t^{2}}dt − 2 ∫_{0}^{∞} e^{−c_{q}λt^{q}−t^{2}}dt.

Then

f′(0) = −2c_{p} ∫_{0}^{∞} t^{p}e^{−t^{2}}dt + 2c_{q} ∫_{0}^{∞} t^{q}e^{−t^{2}}dt = 2 · [c_{q}Γ (\frac{q + 1}{2}) − c_{p}Γ (\frac{p + 1}{2})] > 0.

Since f(0) = 0, it follows that there is some δ = δ_{p,q} > 0 such that for every 0 < λ < δ, f(λ) > 0, i.e. α(p, c_{p}λ) > α(q, c_{q}λ). Part a) of Proposition 17 now implies that μ_{p,c_{p}λ} ≺ μ_{q,c_{q}λ}. As before, tensorization and an application of Lemma 14 give that for every λ < δ,

\frac{𝔼 exp (−^{\frac{λc_{p}∥G∥_{E∩B^{n}_{p}}^{p}}{2^{p/2}}} )}{𝔼 exp (−^{\frac{λc_{p}∥G∥_{B^{k}_{p}}^{p}}{2^{p/2}}} )} ≤ \frac{𝔼 exp (−^{\frac{λc_{q}∥G∥_{E∩B^{n}_{q}}^{q}}{2^{q/2}}} )}{𝔼 exp (−^{\frac{λc_{q}∥G∥_{B^{k}_{q}}^{q}}{2^{q/2}}} )},

and the required inequality follows by taking derivatives at 0 and letting c_{p} and c_{q} tend to 1/Γ (\frac{p+1}{2}) and 1/Γ (\frac{q+1}{2}), respectively.

Remark: Assume that 0 < p < 2. By Proposition 17, for every λ > 0, μ_{p,λ} ≤ μ_{2,λ} = γ̄, where γ̄ has density e^{−πx^{2}} on ℝ. Hence, by rotation invariance of this Gaussian density, one has that for every λ > 0

𝔼e^{−λ∥G∥_{E∩B^{n}_{p}}^{p}} ≤ 𝔼e^{−λ∥G∥_{B^{k}_{p}}^{p}} .

Thus, for any (reasonable) measure τ on [0, ∞),

𝔼 ∫_{0}^{∞} e^{−λ∥G∥_{E∩B^{n}_{p}}^{p}} dτ(λ) ≤ 𝔼 ∫_{0}^{∞} e^{−λ∥G∥_{B^{k}_{p}}^{p}} dτ(λ),

which by Bernstein’s Theorem (see, e.g., the book [32]) implies that for every f : [0, ∞) → ℝ which is completely monotonic,

𝔼f(∥G∥_{E∩B_{p}^{n}}^{p}) ≤ 𝔼f(∥G∥_{B_{p}^{k}}^{p}),

provided these expectations are finite.

Two particular cases which should be singled out are f(t) = e^{−λt^{θ}} for 0 < θ ≤ 1 and λ > 0, and f(t) = t^{−η} for η > 0. The first case implies that for every λ > 0,

𝔼e^{−λ∥G∥_{E∩B^{n}_{p}}^{θp}} ≤ 𝔼e^{−λ∥G∥_{B^{k}_{p}}^{θp}} ,

which, by differentiation at 0 yields

𝔼∥G∥_{E∩B_{p}^{n}}^{θp} ≥ 𝔼∥G∥_{B_{p}^{k}}^{θp}.

From the second case it is evident that for 0 < α < k,

𝔼∥G∥_{E∩B_{p}^{n}}^{−α} ≤ 𝔼∥G∥_{B_{p}^{k}}^{−α}.

The condition α < k is imposed to ensure that these expectations would be finite.

When 2 < p < ∞, γ̄ ≺ μ_{p,λ}, and all the above inequalities are reversed. Summarizing, we obtain

Corollary 19. Let E be a k dimensional subspace of ℝ^{n}. Then for 0 < p < 2 and every 0 < α < k and 0 < β ≤ p,

𝔼∥G∥_{E∩B_{p}^{n}}^{−α} ≤ 𝔼∥G∥_{B_{p}^{k}}^{−α} and 𝔼∥G∥_{E∩B_{p}^{n}}^{β} ≥ 𝔼∥G∥_{B_{p}^{k}}^{β}.

If 2 < p < ∞ then for every 0 < α < k and 0 < β ≤ p,

𝔼∥G∥_{E∩B_{p}^{n}}^{−α} ≥ 𝔼∥G∥_{B_{p}^{k}}^{−α} and 𝔼∥G∥_{E∩B_{p}^{n}}^{β} ≤ 𝔼∥G∥_{B_{p}^{k}}^{β}.

The following proposition is a corollary of parts d) and e) in Proposition 17.

Proposition 20. Let E be a k-dimensional subspace of ℝ^{n}. Then the function

λ ≥ 0 ↦ r_{p}(λ) := \frac{𝔼e^{−λ∥G∥_{E∩B^{n}_{p}}^{p}}}{𝔼e^{−λ∥G∥_{B^{k}_{p}}^{p}}}

is non-increasing when p ≤ 2 and non-decreasing when p ≥ 2.

Remark: Since r_{p}(0) = 1 we have an alternative proof to Corollary 19. Additionally the limit of r_{p}(λ) when λ tends to infinity is

\frac{∫_{E} e^{−∥x∥_{E∩B^{n}_{p}}^{p}} dx}{∫_{ℝ^{k}} e^{−∥x∥_{B^{k}_{p}}^{p}} dx} = \frac{vol_{k}(E ∩ B_{p}^{n})}{vol_{k}(B_{p}^{k})}.

The above equality can be proved by polar integration. The comparison between r_{p}(0) and r_{p}(+∞) yields an alternative proof of the Meyer-Pajor Theorem [19] which uses a different interpolation between exp(−t^{2}) and exp(−|t|^{p}).

3.2 Bounds via convolution inequalities

In this section we derive upper bounds on the Laplace transform of ∥G∥_{E∩B_{p}^{n}}^{p} for p > 2. The main tool is Ball’s version of the Brascamp-Lieb inequality [9, 2]. We follow the method of [2] where the main focus was on the volume of sections.

Let E be a k-dimensional subspace of ℝ^{n} and let P be the orthogonal projection onto E. The canonical basis of ℝ^{n} provides a decomposition of the identity map as ∑_{i=1}^{n} e_{i}⊗e_{i} = Id_{n}, where (v ⊗ v)(x) = ⟨x, v⟩v. Projecting this relation onto E yields a decomposition of the identity on E

∑_{i=1}^{n} P e_{i} ⊗ P e_{i} = Id_{E}.

Setting c_{i} = |P e_{i}|^{2} and u_{i} = P e_{i}/|P e_{i}| (or any unit vector if the norm of P e_{i} is 0), this rewrites as ∑_{i=1}^{n} c_{i}u_{i}⊗u_{i} = Id_{E}. Let λ > 0, and note that for any x ∈ E the i-th coordinate in the canonical basis is x_{i} = ⟨x, e_{i}⟩ = ⟨P x, e_{i}⟩ = ⟨x, P e_{i}⟩ = \sqrt{c_{i}}⟨x, u_{i}⟩. Hence,

∫_{E} e^{−λ∥x∥_{p}^{p}−∥x∥_{2}^{2}/2}dx = ∫_{E} ∏_{i=1}^{n} e^{−λ|x_{i}|^{p}−|x_{i}|^{2}/2}dx

= ∫_{E} ∏_{i=1}^{n} e^{−λ|\sqrt{c_{i}}⟨x,u_{i}⟩|^{p}−c_{i}⟨x,u_{i}⟩^{2}/2}dx

= ∫_{E} ∏_{i=1}^{n} (e^{−λc_{i}^{p/2−1}|⟨x,u_{i}⟩|^{p}−⟨x,u_{i}⟩^{2}/2})^{c_{i}} dx

≤ ∏_{i=1}^{n} (∫_{ℝ} e^{−λc_{i}^{p/2−1}|t|^{p}−t^{2}/2}dt)^{c_{i}}

= exp [∑_{i=1}^{n} c_{i} log ψ (\frac{1}{\sqrt{c_{i}}} )] ,

where we have set ψ(s) = 2 ∫_{0}^{∞} e^{−λs^{2−p}t^{p}−t^{2}/2}dt. First, observe that for p > 2 the function defined on (0, ∞) × [0, ∞) by (s, t) → −λs^{2−p}t^{p} − t^{2}/2 is concave (indeed, on this set the function (s, t) → s^{2−p}t^{p} is convex, as follows from a direct calculation of its Hessian matrix). Therefore, by a well known result of Borell, Prékopa and Rinott (see e.g. [26]), log ψ(s) is a concave function of s > 0 (because it is the integral in t of a log-concave function of (s, t)). Lemma 22 below ensures that the map

s > 0 ↦ s log ψ(\frac{1}{\sqrt{s}})

is concave. This property can be combined with the relation ∑_{i=1}^{n} c_{i} = k (which follows by taking traces in the decomposition of the identity). It yields that for p ≥ 2

∫_{E} e^{−λ∥x∥_{p}^{p}−∥x∥_{2}^{2}/2}dx ≤ (∫_{ℝ} e^{−λ(\sqrt{k/n})^{p−2}|t|^{p}−t^{2}/2}dt)^{k} .

Returning to our previous setting, it implies that for every λ > 0

𝔼e^{−λ∥G∥_{E∩B^{n}_{p}}^{p}} ≤ 𝔼e^{−λ(\sqrt{k/n})^{p−2}∥G∥_{B^{k}_{p}}^{p}} .

Integrating this inequality against positive measures on [0, ∞) and applying Bernstein’s Theorem [32], it follows that for every completely monotonic function f : [0, ∞) → [0, ∞)

𝔼f (λ∥G∥_{E∩B_{p}^{n}}^{p}) ≤ 𝔼f (λ (\sqrt{k/n})^{p−2} ∥G∥_{B_{p}^{k}}^{p}) .

In particular, the following corollary is evident.

Corollary 21. For any p ≥ 2, every 0 ≤ θ ≤ 1 and every λ ≥ 0:

𝔼e^{−λ∥G∥_{E∩B^{n}_{p}}^{θp}} ≤ 𝔼e^{−λ(\sqrt{k/n})^{θ(p−2)}∥G∥_{B^{k}_{p}}^{θp}} .

In particular, by differentiation at 0 it follows that for every 0 ≤ β ≤ p,

𝔼∥G∥_{E∩B_{p}^{n}}^{β} ≥ (\frac{k}{n})^{β( \frac{1}{2} − \frac{1}{p} )} 𝔼∥G∥_{B_{p}^{k}}^{β}.

Also, for every 0 ≤ α < k

𝔼∥G∥_{E∩B_{p}^{n}}^{−α} ≤ (\frac{n}{k})^{α( \frac{1}{2} − \frac{1}{p} )} 𝔼∥G∥_{B_{p}^{k}}^{−α}.

Remark. Assume that k divides n, and write n = mk. Consider the subspace F ⊂ ℝ^{n} which is the “main diagonal” with respect to the decomposition ℝ^{n} = ℝ^{k} × ⋯ × ℝ^{k}, (i.e. F = {(x_{1}, …, x_{m}); x_{i} ∈ ℝ^{k}, x_{1} = … = x_{m}}). Then

𝔼∥G∥_{F ∩B_{p}^{n}}^{p} = m (\frac{1}{\sqrt{m}})^{p} 𝔼∥G∥_{B_{p}^{k}}^{p} = (\frac{k}{n})^{\frac{p}{2} −1} 𝔼∥G∥_{B_{p}^{k}}^{p}

which shows that when k divides n, the case β = p in Corollary 21, is optimal.

Lemma 22. Let c : [0, ∞) → [0, ∞) be a non-decreasing concave function. Then the function f(t) := t c (\frac{1}{\sqrt{t}}), defined for t > 0 is concave.

Proof. We may assume that c is twice continuously differentiable. Clearly

f′(t) = c (\frac{1}{\sqrt{t}}) − \frac{1}{2\sqrt{t}}c′ (\frac{1}{\sqrt{t}}) ,

which is non-increasing provided the function g(u) = c(u) − \frac{u}{2}c′(u) is non-decreasing on [0, ∞). Now, g′(u) = \frac{c′(u)}{2} − \frac{u}{2}c′′(u) is non-negative by our assumptions on c.

3.3 Gaussian measures of sections of the cube

In view of the previous results, one is tempted to conjecture that the following distributional inequality holds for Gaussian measures of sections of dilates of the ℓ_{p}^{n}-ball, that is, for every k-dimensional subspace E and every r > 0, γ_{k}(rB_{p}^{k}) ≤ γ_{E}(E ∩rB_{p}^{n}) if p ≥ 2 and the reverse inequality for p ≤ 2. If such a statement were true, some of the previous results would follow by integration. Unfortunately, it seems that the known techniques are insufficient for this purpose. The product structure of the cube will, however, allow us to prove this conjecture for p = ∞.

By Lemma 13, for every k-dimensional subspace E ⊂ ℝ^{n} and r > 0

γ_{E}(E ∩ rB_{∞}^{n}) = lim_{ϵ→0} (\frac{π}{2ϵ^{2}} )^{\frac{n−k}{2}} \frac{1}{(2π)^{n/2}} ∫_{E(ϵ)} ∏_{i=1}^{n} e^{− \frac{x^{2}_{i}}{2}} 1_{[−r,r]}(x_{i})dm_{n}(x).

Let θ(r) = θ be such that

∫_{−r/θ}^{r/θ} e^{− \frac{θ^{2}t^{2}}{2}} dt = 1,

i.e.,

θ(r) = ∫_{−r}^{r} e^{− \frac{t^{2}}{2}} dt.

We require the following observation: Lemma 23. The function r ↦ θ(r) is increasing and the function r ↦ \frac{θ(r)}{r} is decreasing. Proof. The first assertion is obvious. The second statement follows by differentiating. Indeed,

\frac{d}{dr} \frac{1}{r} ∫_{0}^{r} e^{− \frac{t^{2}}{2}} dt = \frac{1}{r} (e^{− \frac{r^{2}}{2}} − \frac{1}{r} ∫_{0}^{r} e^{− \frac{t^{2}}{2}} dt) = \frac{1}{r^{2}} ∫_{0}^{r} (e^{− \frac{r^{2}}{2}} − e^{− \frac{t^{2}}{2}} ) dt ≤ 0.

Denote by ρ_{r} the probability measure on ℝ defined by

dρ_{r}(t) = e^{− \frac{θ(r)^{2}t^{2}}{2}} 1_{[−r/θ(r),r/θ(r)]}(t)dt.

Thus,

γ_{E}(E ∩ rB_{∞}^{n}) = lim_{ϵ→0} (\frac{π}{2ϵ^{2}} )^{\frac{n−k}{2}} \frac{θ(r)^{n}}{(2π)^{n/2}} ∫_{E( \frac{ϵ}{θ(r)} )} ∏_{i=1}^{n} e^{− \frac{θ(r)^{2}y^{2}_{i}}{2}} 1_{[−r,r]}(θ(r)y_{i})dm_{n}(y)
= lim_{ϵ→0} (\frac{2}{π} )^{k/2} · \frac{θ(r)^{k}}{2^{n}ϵ^{n−k}} · ρ_{r}^{⊗n}(E(ϵ)).

Observe that

γ_{k}(rB_{∞}^{k}) = \frac{1}{(2π)^{k/2}} (∫_{−r}^{r} e^{− \frac{t^{2}}{2}} dt)^{k} = \frac{θ(r)^{k}}{(2π)^{k/2}} ,

hence

\frac{γ_{E}(E ∩ rB_{∞}^{n})}{γ_{k}(rB_{∞}^{k})} = lim_{ϵ→0} \frac{1}{(2ϵ)^{n−k}} · ρ_{r}^{⊗n}(E(ϵ)).  (7)

Lemma 24. For every r > s > 0, ρ_{r} ≺ ρ_{s}. Proof. As usual, define h : [0, ∞) → ℝ by

h(a) = ∫_{0}^{a} [e^{− \frac{θ(r)^{2}t^{2}}{2}} 1_{[−r/θ(r),r/θ(r)]}(t) − e^{− \frac{θ(s)^{2}t^{2}}{2}} 1_{[−s/θ(s),s/θ(s)]}(t)] dt,

and our goal is to show that h(a) ≤ 0 for all a ≥ 0. By Lemma 23, \frac{r}{θ(r)} ≥ \frac{s}{θ(s)}, so that h(a) = 0 for a ≥ \frac{r}{θ(r)}. Moreover, for \frac{s}{θ(s)} ≤ a ≤ \frac{r}{θ(r)}, h(a) = ρ_{r}([0, a]) − 1 ≤ 0. Finally, for 0 ≤ a ≤ \frac{s}{θ(s)},

h(a) = ∫_{0}^{a} [e^{− \frac{θ(r)^{2}t^{2}}{2}} − e^{− \frac{θ(s)^{2}t^{2}}{2}} ] dt ≤ 0,

since θ(r) ≥ θ(s).

By (7), tensorizing the above lemma yields

Theorem 9. For every k-dimensional subspace E ⊂ ℝ^{n} the function:

r ↦ \frac{γ_{E}(E ∩ rB_{∞}^{n})}{γ_{k}(rB_{∞}^{k})} r > 0,

is non-increasing. In particular, by passing to the limit r → ∞ it follows that for every r > 0,

γ_{E}(E ∩ rB_{∞}^{n}) ≥ γ_{k}(rB_{∞}^{k}).

By arguments analogous to those in Section 3.2 one can also obtain the following upper bound on the Gaussian measure of sections of dilates of the cube, which is a Gaussian analog of Ball’s slicing theorem in [2]. As noted in Section 3.2, these bounds are optimal when k divides n.

Theorem 10. For every k-dimensional subspace E ⊂ ℝ^{n} and every r > 0

γ_{E}(E ∩ rB_{∞}^{n}) ≤ γ_{k} (r\sqrt{\frac{n}{k}}B_{∞}^{k}) .

3.4 An application: a remark on the Komlós conjecture

In this section we apply the results of the previous section to prove the following proposition, which was stated in the introduction:

Proposition 25. There is an absolute constant C > 0 such that for every integer m > 0 and every x_{1}, …, x_{m} ∈ ℓ_{∞}, if we denote by d the dimension of the linear span of x_{1}, …, x_{m} then there are signs ε_{1}, …, ε_{m} ∈ {−1, 1} such that

∥∑_{i=1}^{m} ε_{i}x_{i}∥_{∞} ≤ C\sqrt{log d} · max_{1≤1≤m} ∥x_{i}∥_{2} ≤ C\sqrt{log m} · max_{1≤1≤m} ∥x_{i}∥_{2}.

Proof. We may assume that x_{1}, …, x_{m} ∈ ℓ_{2}, in which case, we may write x_{i} = y_{i} + z_{i}, where y_{i} ∈ ℓ_{∞}^{N} for some (large) N, and ∥z_{i}∥_{∞} ≤ 1/m. Denote E = span{y_{1}, …, y_{m}} and let d′ be the dimension of E. There is a constant c > 0 such that for r = c\sqrt{log d′} ≤ c\sqrt{log d}, γ_{d′}(rB_{∞}^{d′}) ≥ \frac{1}{2}. By Theorem 9, if we set K = E ∩ rB_{∞}^{N} then γ_{E}(K) ≥ \frac{1}{2}. By Banaszczyk’s theorem [4], there are signs ε_{1}, …, ε_{m} ∈ {−1, 1} such that ∑_{i=1}^{m} ε_{i}y_{i} ∈ cK, where c is an absolute constant. Hence

∥∑_{i=1}^{m} ε_{i}x_{i}∥_{∞} ≤ ∥∑_{i=1}^{m} ε_{i}y_{i}∥_{∞} + ∑_{i=1}^{m} ∥z_{i}∥_{∞} ≤ (c + 1)\sqrt{log d}.

It is equally simple to deduce the following ℓ_{p} version of this result for p > 2:

Proposition 26. There is an absolute constant C > 0 such that for every 2 ≤ p < ∞, every integer m > 0 and every x_{1}, …, x_{m} ∈ ℓ_{p}, if we denote by d the dimension of the linear span of x_{1}, …, x_{m} then there are signs ε_{1}, …, ε_{m} ∈ {−1, 1} such that

∥∑_{i=1}^{m} ε_{i}x_{i}∥_{p} ≤ C\sqrt{p} · d^{1/p} · max_{1≤1≤m} ∥x_{i}∥_{2} ≤ C\sqrt{p} · m^{1/p} · max_{1≤1≤m} ∥x_{i}∥_{2}.

Proof. As before, we may assume that x_{1}, …, x_{m} ∈ ℓ_{∞}^{N} for some large N. By Corollary 19, if we set E = span{x_{1}, …, x_{m}} then

𝔼∥G∥_{E∩B_{p}^{N}}^{p} ≤ 𝔼∥G∥_{B_{p}^{d}}^{p} = d𝔼|g_{1}|^{p} = O (dp^{p/2}) .

Hence, for every r > 0,

γ_{E}(E ∩ rB_{p}^{N}) = 1 − P (∥G∥_{E∩B_{p}^{N}}^{p} ≥ r^{p}) ≥ 1 − \frac{𝔼∥G∥_{E∩B_{p}^{N}}^{p}}{r^{p}} ≥ 1 − O (\frac{dp^{p/2}}{r^{p}} ) .

Setting K = E ∩ rB_{p}^{N}, then for some r = O (\sqrt{p} · d^{1/p}), γ_{E}(K) ≥ \frac{1}{2}, which concludes the proof by Banaszczyk’s theorem [4].

Remark: The above estimate can actually be improved to give tail estimates as follows. Let E be an m-dimensional subspace of ℝ^{n}. Since for p > 2 the function x ↦ ∥x∥_{p} is Lipschitz with constant 1 on ℝ^{n} and the Gaussian isoperimetric inequality shows that for every ϵ > 0,

γ_{E} (E ∩ (𝔼∥G∥_{E∩B_{p}^{n}} + ϵ)B_{p}^{n}) ≥ 1 − e^{− \frac{ϵ^{2}}{2}} .

Since 𝔼∥G∥_{E∩B_{p}^{n}} ≤ 𝔼∥G∥_{B_{p}^{k}} ≤ c\sqrt{p} · m^{1/p} for some absolute constant c, then

γ_{E} (E ∩ (c\sqrt{p} · m^{1/p} + ϵ)B_{p}^{n}) ≥ 1 − e^{− \frac{ϵ^{2}}{2}} .

3.5 An application: covering numbers of convex hulls of points in ℓ_{2} by B_{p} balls

In this section, which is similar in spirit to the previous one, we use our results to give an infinite dimensional extension of a classical inequality which bounds the minimal number of cubes εB_{∞}^{d} required to cover a convex hull of a finite number of points in ℓ_{2}^{d} (this classical result depends on the maximum of d and the number of points). Here, we are interested in finding upper bounds of the minimal number of cubes εB_{∞} required to cover a convex hull of a finite number of points in ℓ_{2} depending only on ε and the number of taken points. Since the structure of ℓ_{∞} depends deeply of the chosen basis in ℓ_{2}, a simple approximation argument is not enough to obtain our result.

The main result of this section, as described in the introduction, is re-stated below:

Proposition 27. There exists an absolute constant C > 0 such that for every integer m, ε > 0 and 2 ≤ p ≤ ∞, for all x_{1}, …, x_{m} in the unit ball of ℓ_{2},

log N(absconv\{x_{1}, …, x_{m}\}, εB_{p}) ≤ C \frac{log m}{ε^{p/(p−1)}} .

Proof. We first prove the Proposition in the case when p = ∞. Since all x_{i}’s are in B_{2} we can find an integer d so that we can write x_{i} = y_{i} + z_{i} with y_{i} ∈ B_{2}^{d} and ∥z_{i}∥_{∞} < ϵ for all i = 1, …, m. If the absolute convex hull of y_{1}, …, y_{m} can be covered by N translates of ϵB_{∞}^{d} then the absolute convex hull of x_{1}, …, x_{m} can be covered by N translates of 2ϵB_{∞}. So, it is enough to prove the result for the y_{i}’s.

Let T : ℓ_{1}^{m} → ℓ_{2}^{d} defined by T e_{i} = y_{i} for all i = 1, …, m, E = span{y_{1}, …, y_{m}} and G is a Gaussian vector in E. Since ∥x_{i}∥_{2} ≤ 1, then by Sudakov’s inequality [31],

sup_{ε>0} ε\sqrt{log N(T (B_{1}^{m}), ε(B_{2}^{d} ∩ E))} ≤ 𝔼 sup_{i=1,...,m} |⟨G, y_{i}⟩| ≤ C\sqrt{log m}.

Moreover, by the dual Sudakov inequality due to Pajor and Tomczak-Jaegermann [22],

sup_{ε>0} ε\sqrt{log N(B_{2}^{d} ∩ E, εB_{∞}^{d})} ≤ 𝔼∥G∥_{ℓ_{∞}^{d}∩E},

and by Corollary 19, 𝔼∥G∥_{ℓ_{∞}^{d}∩E} ≤ 𝔼∥G∥_{ℓ_{∞}^{dimE}} ≤ C\sqrt{log m}. Therefore,

sup_{ε>0} ε\sqrt{log N(B_{2}^{d} ∩ E, εB_{∞}^{d})} ≤ C\sqrt{log m}.

Since the covering numbers are sub-additive,

log N(T (B_{1}^{m}), εB_{∞}^{d}) ≤ log N(T (B_{1}^{m}), \sqrt{ε}(B_{2}^{d} ∩ E)) + log N(\sqrt{ε}(B_{2}^{d} ∩ E), εB_{∞}^{d}) ≤ C · \frac{log m}{ε}.

For a general p ≥ 2, the proof follows by interpolation. Recall that for Banach spaces X, Y and a compact operator u : X → Y, the entropy numbers of u are defined for every integer k by

e_{k}(u : X → Y) = inf{ε; N(u(B_{X}), εB_{Y}) ≤ 2^{k}}.

Let T be defined as before on ℓ_{1}^{m} by T e_{i} = x_{i} for all i = 1, …, m. It is well known (see lemma 12.1.11 in [24]) that for every integer k,

e_{2k−1}(T : ℓ_{1}^{m} → ℓ_{p}) ≤ e_{k}(T : ℓ_{1}^{m} → ℓ_{2})^{2/p}e_{k}(T : ℓ_{1}^{m} → ℓ_{∞})^{1−2/p}.

The above result for p = ∞, stated in terms of entropy numbers, is

e_{k}(T : ℓ_{1}^{m} → ℓ_{∞}) ≤ C · \frac{log m}{k},

and Sudakov’s inequality [31] is just

e_{k}(T : ℓ_{1}^{m} → ℓ_{2}) ≤ C · \sqrt{\frac{log m}{k}}.

Therefore,

e_{2k−1}(T : ℓ_{1}^{m} → ℓ_{p}) ≤ C · (\frac{log m}{k})^{1−1/p} ,

as claimed.

Acknowledgements: We thank A. Giannopoulos and M. Talagrand for their stimulating questions.

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