The variance conjecture on projections of the cube

TL;DR

None

Contribution

None

Abstract

We prove that the uniform probability measure $\mu$ on every $(n-k)$-dimensional projection of the $n$-dimensional unit cube verifies the variance conjecture with an absolute constant $C$ $$\textrm{Var}_\mu|x|^2\leq C \sup_{\theta\in S^{n-1}}{\mathbb E}_\mu\langle x,\theta\rangle^2{\mathbb E}_\mu|x|^2, $$ provided that $1\leq k\leq\sqrt n$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces.

Equations

$${\text{Var}}_{\mu }|x{|}^2\le C\underset{\theta \in S^{n-1}}{\sup }{\mathbb{E}}_{\mu }⟨x,\theta {⟩}^2{\mathbb{E}}_{\mu }|x{|}^2,$$

$${\text{Var}}_{\mu }|x{|}^2\le C{\lambda }_{\mu }^2{\mathbb{E}}_{\mu }|x{|}^2,$$

$${\text{Var}}_{\mu }g(x)\le C{\lambda }_{\mu }^2{\mathbb{E}}_{\mu }|\nabla g(x){|}^2.$$

$$B_{\infty }^n:=\{x\in {\mathbb{R}}^n:|x_i|\le 1,\forall 1\le i\le n\}$$

$${\text{Var}}_{\mu }|x{|}^2\le C{\lambda }_{\mu }^2{\mathbb{E}}_{\mu }|x{|}^2.$$

$${\text{Var}}_{\mu }|x{|}^2\le C{\lambda }_{\mu }^2{\mathbb{E}}_{\mu }|x{|}^2$$

$${\mathbb{E}}_{\mu }f$$

$$=$$

$$\sum _{i=1}^l\frac{|P_E(F_i)|}{|K|}=1.$$

$$F_{(i_1,{\epsilon }_1,\dots ,i_k,{\epsilon }_k)}=\{x\in B_{\infty }^n:x_{i_j}={\epsilon }_j,j=1,\dots ,k\},$$

$${\text{Var}}_{\mu }|x{|}^2=\sum _{i=1}^l\frac{|P_E(F_i)|}{|K|}{\text{Var}}_{P_E(F_i)}|x{|}^2+\sum _{i=1}^l\frac{|P_E(F_i)|}{|K|}{\left({\mathbb{E}}_{P_E(F_i)}|x{|}^2-{\mathbb{E}}_{\mu }|x{|}^2\right)}^2.$$

$${\mathbb{E}}_{P_E(F)}⟨x,\theta {⟩}^2={\mathbb{E}}_F⟨P_{E}x,\theta {⟩}^2=\frac{1}{3}+{\left(\sum _{j=1}^k{\epsilon }_j{\theta }_{i_j}\right)}^2-\frac{1}{3}\sum _{j=1}^k{\theta }_{i_j}^2$$

$${\mathbb{E}}_{P_E(F)}|x{|}^2={\mathbb{E}}_F|P_{E}x{|}^2=\frac{n-k}{3}+{\left|P_E\left(\sum _{j=1}^k{\epsilon }_j{e}_{i_j}\right)\right|}^2-\frac{1}{3}\sum _{j=1}^k|P_E(e_{i_j}){|}^2.$$

$$\frac{n-2k}{3}\le {\mathbb{E}}_{P_E(F)}|x{|}^2\le \frac{n+2k}{3},\frac{n-2k}{3}\le {\mathbb{E}}_{\mu }|x{|}^2\le \frac{n+2k}{3},$$

$${\lambda }_{\mu }^2\ge \frac{n-2k}{3(n-k)}.$$

$${\mathbb{E}}_F⟨P_{E}x,\theta {⟩}^2$$

$$\frac{1}{n-k}{\mathbb{E}}_F|P_{E}x{|}^{2}dx$$

$$0\le {\left|P_E\left(\sum _{j=1}^k{\epsilon }_j{e}_{i_j}\right)\right|}^2\le {\left|\sum _{j=1}^k{\epsilon }_j{e}_{i_j}\right|}^2=k$$

$$0\le |P_E(e_{i_j}){|}^2\le |e_{i_j}{|}^2=1,$$

$${\lambda }_{\mu }^2\ge {\int }_{S_E}{\mathbb{E}}_{\mu }⟨x,\theta {⟩}^{2}d{\sigma }_E(\theta )=\frac{1}{n-k}{\mathbb{E}}_{\mu }|x{|}^2\ge \frac{n-2k}{3(n-k)},$$

$$P_E(F)=a_F+T_F(B_{\infty }^{n-k})$$

$${\text{Var}}_{{\nu }_a}|x{|}^2={\text{Var}}_{\nu }|x{|}^2+4{\mathbb{E}}_{\nu }⟨a,x{⟩}^2.$$

$$|x+a{|}^2=|x{|}^2+2⟨a,x⟩+|a{|}^2$$

$$|x+a{|}^4=|x{|}^4+4|x{|}^2⟨a,x⟩+4|a{|}^2⟨a,x⟩+2|x{|}^2|a{|}^2+4⟨a,x{⟩}^2+|a{|}^4$$

$$\textrm{Var}_{\nu_{a}}|x|^{2}=\mathbb{E}_{\nu_{a}}|x|^{4}-\big{(}\mathbb{E}_{\nu_{a}}|x|^{2}\big{)}^{2}=\mathbb{E}_{\nu}|x+a|^{4}-\big{(}\mathbb{E}_{\nu}|x+a|^{2}\big{)}^{2}=$$

$$=\mathbb{E}_{\nu}|x+a|^{4}-\big{(}\mathbb{E}_{\nu}|x|^{2}+|a|^{2}\big{)}^{2}=\textrm{Var}_{\nu}|x|^{2}+4\mathbb{E}_{\nu}\langle a,x\rangle^{2}$$

$${\text{Var}}_{P_E(F)}|x{|}^2={\text{Var}}_{T_F(B_{\infty }^{n-k})}|x{|}^2+4{\mathbb{E}}_{T_F(B_{\infty }^{n-k})}⟨a_F,x{⟩}^2$$

$$\displaystyle\mathbb{E}_{T_{F}(B_{\infty}^{n-k})}\langle x,\theta\rangle^{2}=\frac{1}{3}-\frac{1}{3}\sum_{j=1}^{k}\theta_{i_{j}}^{2}\quad\big{(}\leq\frac{1}{3}\big{)}$$

$$\mathbb{E}_{T_{F}(B_{\infty}^{n-k})}|x|^{2}=\frac{n-k}{3}-\frac{1}{3}\sum_{j=1}^{k}|P_{E}(e_{i_{j}})|^{2}\quad\big{(}\leq\frac{n-k}{3}\big{)}$$

$${\mathbb{E}}_{P_E(F)}⟨x,\theta {⟩}^2$$

Full text

[TABLE]

provided that $1\leq k\leq\sqrt{n}$. We also prove that if $1\leq k\leq n^{\frac{2}{3}}(\log n)^{-\frac{1}{3}}$, the conjecture is true for the family of uniform probabilities on its projections on random $(n-k)$-dimensional subspaces.

Partially supported by MINECO Spanish grant MTM2016-77710-P and DGA grant E-64

1. Introduction and notation

The (generalized) variance conjecture states that there exists an absolute constant $C$ such that for every centered log-concave probability $\mu$ on $\mathbb{R}^{n}$ (i.e. of the form $d\mu=e^{-v(x)}dx$ for some convex function $v:\mathbb{R}^{n}\to(-\infty,\infty]$)

[TABLE]

where $\mathbb{E}_{\mu}$ and $\textrm{Var}_{\mu}$ denote the expectation and the variance with respect to $\mu$ and $\lambda_{\mu}$ is the largest eigenvalue of the covariance matrix, i.e. $\lambda_{\mu}^{2}=\max_{\theta\in S^{n-1}}\mathbb{E}_{\mu}\langle x,\theta\rangle^{2}$ where $S^{n-1}$ denotes the unit Euclidean sphere in $\mathbb{R}^{n}$.

This conjecture was first considered in the context of the so called Central Limit Problem for isotropic convex bodies in [BK] and it is a particular case of a more general statement, known as the Kannan, Lovász, and Simonovits or KLS-conjecture, see [KLS], which conjectures the existence of an absolute constant $C$ such that for any centered log-concave probability in $\mathbb{R}^{n}$ and any locally Lipschitz function $g:\mathbb{R}^{n}\to\mathbb{R}$ such that $\textrm{Var}_{\mu}g(x)$ is finite

[TABLE]

In recent years a number of families of measures have been proved to verify these conjectures (see [AB2] for a recent review on the subject). For instance, the KLS-conjecture is known to be true for the Gaussian probability and the uniform probability measures on the $\ell_{p}^{n}$-balls, some revolution bodies, the simplex and, with an extra $\log n$ factor, on unconditional bodies and log-concave probabilities with many symmetries (see [BaC], [BaW], [B], [H], [K], [LW], [S]). The best general known result for the KLS-conjecture adds a factor $\sqrt{n}$ and is due to Lee and Vempala (see [LV]). Besides, the variance conjecture is known to be true for uniform probabilities on unconditional bodies and on hyperplane projections of the cross-polytope and the cube (see [K] and [AB1]). The best general estimate for the variance conjecture is the one given by Lee and Vempala for the KLS-conjecture.

We would like to remark that, while in the case of the KLS-conjecture one can assume without loss of generality that $\mu$ is isotropic (since then every linear transformation of the measure verifies it) this is not the case when we restrict to the variance conjecture, as we are considering only the function $g(x)=|x|^{2}$.

Before stating our results let us introduce some more notation. Let

[TABLE]

denote the $n$-dimensional unit cube and, for any $1\leq k\leq n$, let $G_{n,n-k}$ be the set of all $(n-k)$-dimensional subspaces of $\mathbb{R}^{n}$. For any $E\in G_{n,n-k}$ we will denote by $K:=P_{E}B_{\infty}^{n}$ the orthogonal projection of $B_{\infty}^{n}$ onto $E$ and by $\mu$ the uniform probability on $K$. As mentioned before, it was proved in [AB1] that the family of uniform probabilities on any $(n-1)$-dimensional projection of $B_{\infty}^{n}$ verifies the variance conjecture.

In this paper we will prove the following

Theorem 1.1**.**

There exists an absolute constant $C$ such that for any $1\leq k\leq\sqrt{n}$ and any $E\in G_{n,n-k}$, if $\mu$ denotes the uniform probability measure on $K=P_{E}B_{\infty}^{n}$, then

[TABLE]

We will also prove the following theorem, which shows that for $k$ in a larger range, the variance conjecture is true for the family of uniform probabilities on the projections of $B_{\infty}^{n}$ on a random $(n-k)$-dimensional subspace. For that matter, we denote by $\mu_{n,n-k}$ the Haar probability measure on $G_{n,n-k}$.

Theorem 1.2**.**

There exist absolute constants $C,c_{1},c_{2}$ such that for any $1\leq k\leq\frac{n^{\frac{2}{3}}}{(\log n)^{\frac{1}{3}}}$, if $\mu$ denotes the uniform probability measure on $K=P_{E}(B_{\infty}^{n})$, the measure $\mu_{n,n-k}$ of the subspaces $E\in G_{n,n-k}$ for which

[TABLE]

is greater than $1-c_{1}e^{-c_{2}n^{\frac{2}{3}}(\log n)^{\frac{2}{3}}}$.

The main tool to prove both theorems will be to decompose an integral on $K$ as the sum of the integrals on the projections of some $(n-k)$-dimensional faces. It was proved in [ABBW] that for any $E\in G_{n,n-k}$ there exist $F_{1},\dots,F_{l}$ a set of $(n-k)$-dimensional faces of $B_{\infty}^{n}$ such that for any integrable function on $K$

[TABLE]

where we have denoted by $|\cdot|$ the relative volume of a convex body to the affine subspace in which it lies, $\mathbb{E}_{F_{i}}$ and by $\mathbb{E}_{P_{E}(F_{i})}$ the expectation with respect to the uniform probability on the face $F_{i}$ and on its projection $P_{E}(F_{i})$. In particular

[TABLE]

Notice that the $(n-k)$-dimensional faces of $B_{\infty}^{n}$ are the sets of the form

[TABLE]

where $1\leq i_{j}\leq n$, $i_{j_{1}}\neq i_{j_{2}}$ and $\varepsilon_{j}=\pm 1$.

For any $E\in G_{n,n-k}$ we write $S_{E}=S^{n-1}\cap E$ and denote by $\sigma_{E}$ the Haar probability measure on $S_{E}$.

2. The variance conjecture on $(n-k)$-dimensional projections of the cube

In this section we shall prove Theorem 1.1. We start with the following lemma, which can be proved by direct computation:

Lemma 2.1**.**

Let $E\in G_{n,n-k}$, $\mu$ the uniform probability on $K=P_{E}(B_{\infty}^{n})$ and $\{F_{i}\}_{i=1}^{l}$ the set of $(n-k)$-dimensional faces described in (1). Then

[TABLE]

We will estimate the two summands appearing in (5). The following lemma provides upper and lower bounds to some of parameters involved.

Lemma 2.2**.**

Let $E\in G_{n,n-k}$. Then, for any $\theta\in S_{E}$ and any $(n-k)$-dimensional face $F=F_{(i_{1},\varepsilon_{1},\dots,i_{k},\varepsilon_{k})}$ of $B_{\infty}^{n}$ we have,

[TABLE]

and

[TABLE]

Consequently,

[TABLE]

and

[TABLE]

Proof.

For every $\theta\in S_{E}$, straightforward computations yield

[TABLE]

This proves the first identity. Now, by integrating on $\theta\in S_{E}$ with respect to the uniform probability mealonsodsure and using Fubini’s theorem, we obtain

[TABLE]

which proves the second identity.

The bounds

[TABLE]

and

[TABLE]

prove the upper and lower bound for $\mathbb{E}_{F}|P_{E}x|^{2}$ and by using formula (1) we deduce the estimates for $\mathbb{E}_{\mu}|x|^{2}$. Finally, notice that

[TABLE]

which proves the last inequality. ∎

We now focus on the first summand in (5). We take into account the fact that for any $(n-k)$-dimensional face $F=F_{(i_{1},\varepsilon_{1},\dots,i_{k},\varepsilon_{k})}$ we can write

[TABLE]

where $a_{F}=P_{E}\left(\sum_{j=1}^{k}\varepsilon_{j}e_{i_{j}}\right)$ and $T_{F}\,:\,\mathbb{R}^{n-k}\to E$ is a linear map.

The effect of the translation map in our problem is the content of the next

Lemma 2.3**.**

Let $\nu$ be a symmetric measure in $\mathbb{R}^{n}$, $a\in\mathbb{R}^{n}$ and $\nu_{a}$ the translate measure $\nu_{a}(A):=\nu(A-a)$ (or equivalently, $\int_{\mathbb{R}^{n}}f(x)\ d\nu_{a}(x)=\int_{\mathbb{R}^{n}}f(x+a)\ d\nu(x)$ for any non negative (measurable) function $f$). Then,

[TABLE]

Proof.

[TABLE]

and

[TABLE]

Taking expectations and using symmetry we have,

[TABLE]

[TABLE]

∎

Taking in the previous lemma $\nu$ as the uniform probability measure on $T_{F}(B_{\infty}^{n-k})$ we have

Corollary 2.1**.**

Let $F=F_{(i_{1},\varepsilon_{1},\dots,i_{k},\varepsilon_{k})}$ be an $(n-k)$-dimensional face of $B_{\infty}^{n}$ and let $P_{E}(F)=a_{F}+T_{F}(B_{\infty}^{n-k})$ as above. Then

[TABLE]

Lemma 2.4**.**

Let $E\in G_{n,n-k}$. Then, for any $\theta\in S_{E}$ and any $(n-k)$-dimensional face of $B_{\infty}^{n}$, $F=F_{(i_{1},\varepsilon_{1},\dots,i_{k},\varepsilon_{k})}$, if $P_{E}(F)=a_{F}+T_{F}(B_{\infty}^{n-k})$ as above, we have

[TABLE]

and

[TABLE]

Proof.

Notice that

[TABLE]

On the other hand, by Lemma 2.2

[TABLE]

and we obtain the result. By integrating in $\theta\in S_{E}$ with respect to the uniform measure and using Fubini’s theorem we obtain the second identity. ∎

As a consequence we have the following lemma, which gives an upper bound for the first term in (5) of the right order for the variance conjecture to be true as long as $k\leq\frac{n}{3}$.

Lemma 2.5**.**

Let $E\in G_{n,n-k}$. Then, for any $(n-k)$-dimensional face $F$ of $B_{\infty}^{n}$ we have,

[TABLE]

Consequently, there exists an absolute constant $C$ such that if $k\leq\frac{n}{3}$ and $\{F_{i}\}_{i=1}^{l}$ is the set of $(n-k)$-dimensional faces described in (1) then

[TABLE]

Proof.

By Corollary 2.1, we have that for any such $F$

[TABLE]

Since $B_{\infty}^{n-k}$ verifies the Kannan-Lovász-Simonovits conjecture, every linear transform of it verifies the variance conjecture and therefore there exists an absolute constant $C$ such that

[TABLE]

Since by Lemma 2.4 the two factors involved are bounded by $\frac{1}{3}$ and $\frac{n-k}{3}$ respectively, we have $\textrm{Var}_{T_{F}(B_{\infty}^{n-k})}|x|^{2}\leq C(n-k)$.

On the other hand, by Lemma 2.4,

[TABLE]

Therefore, there exists an absolute constant $C$ such that

[TABLE]

which proves the first part of the Lemma.

For the second part, notice that by Lemma 2.2 we have

[TABLE]

when $1\leq k\leq\frac{n}{3}$ and now the second part of the Lemma easily follows. ∎

For the second summand of (5) we invoke once again Lemma 2.2. The estimates therein provide an upper bound of the right order for the variance conjecture to hold as long as $k\leq\sqrt{n}$.

Lemma 2.6**.**

Let $E\in G_{n,n-k}$ and let $\mu$ be the uniform probability on $K=P_{E}(B_{\infty}^{n})$. Then for any $(n-k)$-dimensional face $F$ of $B_{\infty}^{n}$ we have,

[TABLE]

Consequently, there exists an absolute constant $C$ such that if $k\leq\sqrt{n}$ and $\{F_{i}\}_{i=1}^{l}$ is the set of $(n-k)$-dimensional faces described in (1),

[TABLE]

Proof.

By Lemma 2.2 we have that for any $(n-k)$ dimensional face $F$

[TABLE]

Therefore,

[TABLE]

On the other hand, using as above the bound $\lambda_{\mu}^{2}\mathbb{E}_{\mu}|x|^{2}\geq\frac{n}{54}$ we have that if $k\leq\sqrt{n}$

[TABLE]

∎

Lemmas 2.5 and 2.6, together with formula (5) prove Theorem 1.1.

3. The variance conjecture on random $(n-k)$-dimensional projections of the cube

We will show that we can improve the range of the codimension $k$ for which the variance conjecture remains true on a random subspace $E\in G_{n,n-k}$. In order to do that we will consider, for any $(n-k)$-dimensional face $F$ of $B_{\infty}^{n}$, the function $f:G_{n,n-k}\to\mathbb{R}$ given by $f(E)=\mathbb{E}_{F}|P_{E}x|^{2}$ and make use of the concentration of measure theorem, proved by Gromov and Milman, on $G_{n,n-k}$ (see, for instance, [MS]). $O(n)$ denotes the orthogonal group equipped with the Hilbert-Schmidt distance $\|\cdot\|_{HS}$ and we represent any $U\in O(n)$ by $U=(u_{1},\dots,u_{n})$, where $(u_{i})$ is an orthonormal basis of $\mathbb{R}^{n}$.

Theorem 3.1** (Concentration of measure).**

Let $f\,:\,G_{n,n-k}\to\mathbb{R}$ be a Lipschitz function with Lipschitz constant $\sigma$ with respect to the distance

[TABLE]

Then, for every $\lambda>0$

[TABLE]

where $c_{1}$ and $c_{2}$ are positive absolute constants.

In the following lemma we compute the expected value of $f$. Let us point out that what matters to us for our purposes is that, due to the symmetries of $B_{\infty}^{n}$, its value does not depend on the face $F$. Nevertheless, we compute its exact value.

Lemma 3.1**.**

Let $F$ be an $(n-k)$-dimensional face of $B_{\infty}^{n}$. Then

[TABLE]

Proof.

Notice that, by Fubini’s theorem and the uniqueness of the Haar measure $\sigma$ on $S^{n-1}$ we have

[TABLE]

∎

In the following lemma we estimate the Lipschitz constant of $f$ with respect to the distance defined in Theorem 3.1. Notice that, as before, its value does not depend on $F$.

Lemma 3.2**.**

Let $F=F_{(i_{1},\varepsilon_{1},\dots,i_{k},\varepsilon_{k})}$ be an $(n-k)$-dimensional face of $B_{\infty}^{n}$ and let $f\,:\,G_{n,n-k}\to\mathbb{R}$ be the function $f(E)=\mathbb{E}_{F}|P_{E}x|^{2}$. For any $E_{1},E_{2}\in G_{n,n-k}$ we have

[TABLE]

Proof.

Let $E_{1},E_{2}\in G_{n,n-k}$. By Lemma 2.2 we have

[TABLE]

Notice that for any $U,V\in O(n)$ such that $E_{1}=\textrm{span}\{u_{1},\dots,u_{n-k}\}$ and $E_{2}=\textrm{span}\{v_{1},\dots,v_{n-k}\}$ we can write $P_{E_{1}}=\sum_{j=1}^{n-k}u_{j}\otimes u_{j}$ and $P_{E_{2}}=\sum_{j=1}^{n-k}v_{j}\otimes v_{j}$, and then, for any such $U,V$

[TABLE]

since $1-\langle u_{j},v_{j}\rangle^{2}\leq 2(1-\langle u_{j},v_{j}\rangle)=|u_{j}-v_{j}|^{2}$. Consequently $\|P_{E_{1}}-P_{E_{2}}\|\leq\sqrt{2}d(E_{1},E_{2})$ and we obtain the result. ∎

Lemma 3.3**.**

Let $1\leq k\leq\frac{n^{\frac{2}{3}}}{(\log n)^{\frac{1}{3}}}$. There exist positive absolute constants $C,c_{1},c_{2}$ such that the set

[TABLE]

has measure $\mu_{n,n-k}$ smaller than $c_{1}e^{-c_{2}n^{\frac{2}{3}}(\log n)^{\frac{2}{3}}}$.

Proof.

Let $F$ be a fixed $(n-k)$-dimensional face of $B_{\infty}^{n}$. Then, taking $\lambda=C\sqrt{n}$ we obtain, using Theorem 3.1 that

[TABLE]

Since the number of $(n-k)$-dimensional faces of $B_{\infty}^{n}$ equals $2^{k}\binom{n}{k}$, using the union bound we have that for any $C>0$

[TABLE]

if we choose $C$ a constant big enough. ∎

As a consequence, we obtain the following lemma, which gives an estimate of the right order for most subspaces, for the second term in (5).

Lemma 3.4**.**

There exists an absolute constant $C$ such that for any $1\leq k\leq\frac{n^{\frac{2}{3}}}{(\log n)^{\frac{1}{3}}}$, if $\mu$ denotes the uniform probability measure on $K=P_{E}(B_{\infty}^{n})$, the measure $\mu_{n,n-k}$ of the subspaces $E\in G_{n,n-k}$ for which

[TABLE]

for every $(n-k)$-dimensional face $F$ of $B_{\infty}^{n}$ is greater than $1-c_{1}e^{-Cn^{\frac{2}{3}}(\log n)^{\frac{2}{3}}}$.

Proof.

By Lemma 3.3 there exists a set of subspaces $\mu_{n,n-k}$ measure greater than $1-c_{1}e^{-Cn^{\frac{2}{3}}(\log n)^{\frac{2}{3}}}$ such that for every $(n-k)$-dimensional face $F$ of $B_{\infty}^{n}$,

[TABLE]

and then for every $F_{1},F_{2}$, $(n-k)$-dimensional faces

[TABLE]

Consequently, since $\displaystyle{\mathbb{E}_{\mu}|x|^{2}=\sum_{i=1}^{l}\frac{|P_{E}(F_{i})|}{|K|}\mathbb{E}_{P_{E}{F_{i}}}|x|^{2}}$, we have that for every $E$ in this set and every face $F$

[TABLE]

and

[TABLE]

∎

Now we are able to prove Theorem 1.2.

Proof of Theorem 1.2.

If $1\leq k\leq\frac{n^{\frac{2}{3}}}{(\log n)^{\frac{1}{3}}}$ by Lemma 2.2 we have $\lambda_{\mu}^{2}\mathbb{E}_{\mu}|x|^{2}\geq Cn$. By equation (5), if $\{F_{i}\}_{i=1}^{l}$ are the $(n-k)$-dimensional faces of $B_{\infty}^{n}$ described in $(1)$ we have,

[TABLE]

By Lemma 2.5 the first maximum is bounded from above by $Cn$ and by Lemma 3.4 there exists a set of $(n-k)$-dimensional subspaces with measure larger than $1-c_{1}e^{-Cn^{\frac{2}{3}}(\log n)^{\frac{2}{3}}}$ such that the second maximum is bounded from above by $Cn$, which proves Theorem 1.2. ∎