CUBE

DAVID ALONSO-GUTI ´ERREZ AND JULIO BERNU ´ES

Abstract. We prove that the uniform probability measureµon every (n− k)-dimensional projection of the n-dimensional unit cube verifies the variance conjecture with an absolute constantC

Varµ|x|2_{≤}_{C} _{sup}
θ∈Sn−1E

µhx, θi2

Eµ|x|2,

provided that 1≤k≤√n. We also prove that if 1≤k≤n23(logn)−13, the conjecture is true for the family of uniform probabilities on its projections on random (n−k)-dimensional subspaces.

1. Introduction and notation

The (generalized) variance conjecture states that there exists an absolute
con-stant C such that for every centered log-concave probability µ on_{R}n _{(i.e. of the}

formdµ=e−v(x)_{dx}_{for some convex function}_{v}_{:}

Rn→(−∞,∞])

Varµ|x|2≤Cλ2µEµ|x|2,

whereEµ and Varµ denote the expectation and the variance with respect toµand

λµis the largest eigenvalue of the covariance matrix, i.e. λ2µ= maxθ∈Sn−1_{E}µhx, θi2

whereSn−1 _{denotes the unit Euclidean sphere in}
Rn.

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 gen-eral statement, known as the Kannan, Lov´asz, and Simonovits or KLS-conjecture, see [KLS], which conjectures the existence of an absolute constant C such that for any centered log-concave probability inRn and any locally Lipschitz function

g:Rn→Rsuch that Varµg(x) is finite

Varµg(x)≤Cλ2µEµ|∇g(x)|2.

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`n

p-balls, some revolution bodies, the simplex and, with

an extra logn 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√nand is due to Lee and Vem-pala (see [LV]). Besides, the variance conjecture is known to be true for uniform

Date: May 24, 2017.

2010Mathematics Subject Classification. Primary 52B09, Secondary 52A23.

Key words and phrases. Variance conjecture, log-concave measures, convex bodies. Partially supported by MINECO Spanish grant MTM2016-77710-P and DGA grant E-64.

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 µ 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 functiong(x) =|x|2_{.}

Before stating our results let us introduce some more notation. Let
B_{∞}n :={x∈_{R}n _{:} _{|}_{x}

i| ≤1,∀1≤i≤n}

denote then-dimensional unit cube and, for any 1≤k≤n, letGn,n−k be the set

of all (n−k)-dimensional subspaces of _{R}n_{. For any} _{E} _{∈} _{G}

n,n−k we will denote

by K := PEB∞n the orthogonal projection of B∞n onto E and by µ the uniform

probability on K. {ei}ni=1 will denote the standard canonical basis in Rn. As

mentioned before, it was proved in [AB1] that the family of uniform probabilities on any (n−1)-dimensional projection ofBn

∞ verifies the variance conjecture.

In this paper we will prove the following

Theorem 1.1. There exists an absolute constantC such that for any1≤k≤√n

and anyE∈Gn,n−k, ifµdenotes the uniform probability measure onK=PEBn∞,

then

Varµ|x|2≤Cλ2µEµ|x|2.

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 ofBn

∞on a random (n−k)-dimensional subspace. For that matter, we

denote byµn,n−k the Haar probability measure onGn,n−k.

Theorem 1.2. There exist absolute constants C, c1, c2 such that for any 1≤k≤

n23

(logn)13

, ifµdenotes the uniform probability measure onK=PE(B∞n), the measure

µn,n−k of the subspaces E∈Gn,n−k for which Varµ|x|2≤Cλ2µEµ|x|2 is greater than 1−c1e−c2n

2

3(logn)23_{.}

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 anyE ∈Gn,n−k there exist F1, . . . , Fl a set of

(n−k)-dimensional faces of Bn

∞ such that for any integrable functionf onK

Eµf := 1 |K| Z K f(x)dx= l X i=1 |PE(Fi)| |K| EPE(Fi)f(x) = l X i=1 |PE(Fi)| |K| 1 |PE(Fi)| Z PE(Fi) f(x)dx = l X i=1 |PE(Fi)| |K| 1 |Fi| Z Fi f(PEx)dx = l X i=1 |PE(Fi)| |K| EFif(PEx), (1)

where we have denoted by| · | the relative volume of a convex body to the affine subspace in which it lies, EFi and by EPE(Fi) the expectation with respect to the

uniform probability on the faceFi and on its projectionPE(Fi). In particular l

X

i=1

|PE(Fi)|

|K| = 1.

In this way, Varµ|x|2 is written as the sum of two averages on some faces (see

Lemma 2.1 below). The restrictions on the range ofkin both theorems arise from the fact that in order to give estimates for these averages we do it for each face of the cube separately. With this method we provide an upper bound for Varµ|x|2

and a lower bound forλ2

µEµ|x|2of the same order. It would be interesting to know

whether one could bound these two terms from above directly byλ2

µEµ|x|2.

Notice that the (n−k)-dimensional faces of B_{∞}n are the sets of the form
F(i1,ε1,...,ik,εk)={x∈B

n

∞ : xij =εj, j= 1, . . . , k},

where 1 ≤ij ≤n, ij1 6=ij2 and εj =±1. In other words, if we divide{1, . . . , n} into two disjoint sets {ij}kj=1 and {lj}nj=1−k of cardinality k and n−k, then the

(n−k)-dimensional face F(i1,ε1,...,ik,εk) is

(2) F(i1,ε1,...,ik,εk)= k X j=1 εjeij +IF(B n−k ∞ ),

whereIF is the linear map fromRn−k to Rn given byIFx=P n−k j=1 xjelj.

For any E ∈ Gn,n−k we write SE = Sn−1∩E and denote by σE the Haar

probability measure onSE.

2. _{The variance conjecture on} (n−k)_{-dimensional projections of the}

cube

In this section we shall prove Theorem 1.1. We will use the aforementioned
representation of _{E}µf in order to write Varµ|x|2 as the sum of two terms. One

of them will be an average of the variances of|x|2 _{on the projections of the faces}

involved in (1). The other one will be an average of the distances from the expected
value of|x|2 _{on the projections of such faces to}

Eµ|x|2. We will give estimates for

these terms valid for every face and prove

Varµ|x|2≤Cn≤Cλ2µEµ|x|2.

We start with the following lemma, which can be proved by direct computation: Lemma 2.1. Let E ∈ Gn,n−k, µ the uniform probability on K = PE(B∞n) and {Fi}li=1 the set of (n−k)-dimensional faces described in (1). Then

(3) Varµ|x|2=
l
X
i=1
|PE(Fi)|
|K| VarPE(Fi)|x|
2_{+}
l
X
i=1
|PE(Fi)|
|K| EPE(Fi)|x|
2_{−}
Eµ|x|2
2
.

We will estimate the two summands appearing in (3). The following lemma provides upper and lower bounds to some of parameters involved.

Lemma 2.2. Let E∈Gn,n−k. Then, for anyθ∈SE and any(n−k)-dimensional
face F=F(i1,ε1,...,ik,εk) ofB
n
∞ we have,
EPE(F)hx, θi
2_{=}
EFhPEx, θi2=
1
3 +
k
X
j=1
εjθij
2
−1
3
k
X
j=1
θi2j
and
EPE(F)|x|2=EF|PEx|2=
n−k
3 +
PE
k
X
j=1
εjeij
2
−1
3
k
X
j=1
|PE(eij)|
2_{.}
Consequently,
n−2k
3 ≤EPE(F)|x|
2_{≤}n+ 2k
3 ,
n−2k
3 ≤Eµ|x|
2_{≤} n+ 2k
3 ,
and
λ2_{µ}≥ n−2k
3(n−k).

Proof. Using (2), we have a random vector x uniformly distributed in F can be written as x= k X j=1 εjeij +IF(y)

wherey is uniformly distributed inB_{∞}n−k. Then, for everyθ∈SE, straightforward

computations yield
EFhPEx, θi2 = EFhx, θi2
= 1
|Bn∞−k|
Z
B∞n−k
k
X
j=1
εjθij+
n−k
X
j=1
yjθlj
2
dy
=
k
X
j=1
εjθij
2
+1
3
n−k
X
j=1
θ2_{l}
j
=
k
X
j=1
εjθij
2
+1
3
1−
k
X
j=1
θ2ij
.

This proves the first identity. Now, by integrating onθ ∈SE with respect to the

uniform probability measure onSE, using the fact that for everyx∈Rn
|PEx|2= (n−k)
Z
SE
hPEx, θi2dσE(θ) = (n−k)
Z
SE
hx, θi2_{dσ}
E(θ),

and using Fubini’s theorem, we obtain
1
n−kEF|PEx|
2 _{=} 1
3+
1
n−k
PE
k
X
j=1
εjeij
2
− 1
3(n−k)
k
X
j=1
|PE(eij)|
2_{,}
sincePk
j=1εjθij
2
=DPk
j=1εjeij, θ
E2
andθ2
ij =heij, θi

2_{. This proves the second}

The bounds
0≤
PE
k
X
j=1
εjeij
2
≤
k
X
j=1
εjeij
2
=k
and
0≤ |PE(eij)|
2
≤ |eij|
2_{= 1,}

prove the upper and lower bound forEF|PEx|2and by using formula (1) we deduce

the estimates forEµ|x|2. Finally, notice that

λ2_{µ}≥
Z
SE
Eµhx, θi2 dσE(θ) =
1
n−kEµ|x|
2_{≥} n−2k
3(n−k),

which proves the last inequality. _{}

We now focus on the first summand in (3). We take into account the fact that for any (n−k)-dimensional face F =F(i1,ε1,...,ik,εk)we can write

PE(F) =aF+TF(B∞n−k), where aF =PE Pk j=1εjeij

andTF : Rn−k →E is the linear map obtained as

the composition ofIF in (2) andPE.

The effect of the translation map in our problem is the content of the next Lemma 2.3. Let ν be a symmetric measure inRn (i.e., ν(A) =ν(−A) for every measurable set A), a ∈ Rn and νa the translate measure νa(A) := ν(A−a) (or equivalently, R

Rnf(x) dνa(x) = R

Rnf(x+a) dν(x) for any integrable measurable
function f). Then,
Varνa|x|
2_{=}_{Var}
ν|x|2+ 4Eνha, xi2.
Proof.
|x+a|2_{=}_{|}_{x}_{|}2_{+ 2}_{h}_{a, x}_{i}_{+}_{|}_{a}_{|}2
and
|x+a|4_{=}_{|}_{x}_{|}4_{+ 4}_{|}_{x}_{|}2_{h}_{a, x}_{i}_{+ 4}_{|}_{a}_{|}2_{h}_{a, x}_{i}_{+ 2}_{|}_{x}_{|}2_{|}_{a}_{|}2_{+ 4}_{h}_{a, x}_{i}2_{+}_{|}_{a}_{|}4_{.}

Taking expectations and using symmetry we have,
Varνa|x|
2_{=}
Eνa|x|
4_{−}
Eνa|x|
22
=Eν|x+a|4− Eν|x+a|2
2
=
=_{E}ν|x+a|4− Eν|x|2+|a|2
2
= Varν|x|2+ 4Eνha, xi2.

Taking in the previous lemmaνas the uniform probability measure onTF(B∞n−k)

we have

Corollary 2.1. LetF =F(i1,ε1,...,ik,εk)be an (n−k)-dimensional face ofB

n

∞ and

letPE(F) =aF+TF(B∞n−k)as above. Then

VarPE(F)|x|2=VarTF(Bn∞−k)|x|

2_{+ 4}

ETF(Bn∞−k)haF, xi

Lemma 2.4. Let E∈Gn,n−k. Then, for anyθ∈SE and any(n−k)-dimensional
face ofBn
∞,F =F(i1,ε1,...,ik,εk), ifPE(F) =aF+TF(B
n−k
∞ )as above, we have
ETF(B∞n−k)hx, θi
2_{=}1
3 −
1
3
k
X
j=1
θ_{i}2
j ≤
1
3
,
and
ETF(Bn∞−k)|x|
2_{=}n−k
3 −
1
3
k
X
j=1
|PE(eij)|
2 _{≤}n−k
3
.

Proof. Notice that

EPE(F)hx, θi
2 _{=}
ETF(B∞n)haF +x, θi
2_{=}_{h}_{a}
F, θi2+ETF(B∞n)hx, θi
2
=
k
X
j=1
εjθij
2
+_{E}TF(B∞n)hx, θi
2_{.}

On the other hand, by Lemma 2.2

EPE(F)hx, θi2=EFhPEx, θi2=
1
3+
k
X
j=1
εjθij
2
−1
3
k
X
j=1
θ_{i}2_{j},

and we obtain the result. By integrating in θ ∈ SE 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 (3) of the right order for the variance conjecture to be true as long
ask≤n

3.

Lemma 2.5. Let E ∈Gn,n−k. Then, for any (n−k)-dimensional face F of B∞n

we have,

VarPE(F)|x|2≤Cn.

Consequently, there exists an absolute constantC such that ifk ≤ n

3 and{Fi}

l i=1

is the set of (n−k)-dimensional faces described in (1) then
l
X
i=1
|PE(Fi)|
|K| VarPE(Fi)|x|
2_{≤}_{Cλ}2
µEµ|x|2.
Proof. By Corollary 2.1, we have that for any suchF

VarF|PEx|2= Var_{T}

F(B∞n−k)|x|

2_{+ 4}

ETF(B∞n−k)haF, xi

2_{.}

Since B_{∞}n−k verifies the Kannan-Lov´asz-Simonovits conjecture (see [LW]), every
linear transform of it verifies the variance conjecture and therefore there exists an
absolute constantC such that

Var_{T}

F(B∞n−k)|x|

2_{≤}_{Cλ}2

TF(Bn∞−k)ETF(B∞n−k)|x|

2_{.}

Since by Lemma 2.4 the two factors involved are bounded by 1_{3}andn−_{3}k respectively,
we have Var_{T}

F(B∞n−k)|x|

2_{≤}_{C(n}_{−}_{k).}

ETF(Bn∞−k)haF, xi
2_{≤} 1
3|aF|
2_{=} 1
3
PE
k
X
j=1
εjeij
2
≤ 1
3
k
X
j=1
εjeij
2
= 1
3k.
Therefore, there exists an absolute constantC such that

VarPE(F)|x|

2_{≤}_{C(n}_{−}_{k}_{+}_{k) =}_{Cn,}

which proves the first part of the Lemma.

For the second part, notice that by Lemma 2.2 we have
λ2_{µ}_{E}µ|x|2≥
(n−2k)2
9(n−k) ≥
n
54
when 1≤k≤n

3 and now the second part of the Lemma easily follows.

For the second summand of (3) 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 ask≤√n.

Lemma 2.6. LetE∈Gn,n−kand letµbe the uniform probability onK=PE(B∞n).

Then for any(n−k)-dimensional faceF ofBn_{∞} we have,
EF|PEx|2−Eµ|x|2
≤
4k
3 .

Consequently, there exists an absolute constantC such that ifk≤√nand{Fi}li=1

is the set of (n−k)-dimensional faces described in (1),
l
X
i=1
|PE(Fi)|
|K| EFi|PEx|
2_{−}
Eµ|x|2
2
≤Cλ2µEµ|x|2.
Proof. By Lemma 2.2 we have that for any (n−k) dimensional faceF

−4k
3 ≤EF|PEx|
2_{−}
Eµ|x|2≤
4k
3 .
Therefore,
l
X
i=1
|PE(Fi)|
|K| EFi|PEx|
2
−Eµ|x|2
2
≤16k
2
9 .
On the other hand, using as above the boundλ2

µEµ|x|2≥ _{54}n we have that ifk≤
√
n
l
X
i=1
|PE(Fi)|
|K| EFi|PEx|
2_{−}
Eµ|x|2
2
≤Cλ2_{µ}_{E}µ|x|2.

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

We will show that we can improve the range of the codimensionkfor which the variance conjecture remains true on a random subspace E ∈Gn,n−k. In order to

do that we will consider, for any (n−k)-dimensional face F of Bn

∞, the function

f :Gn,n−k →Rgiven by f(E) =EF|PEx|2 and make use of the concentration of

measure theorem, proved by Gromov and Milman, on Gn,n−k (see, for instance,

[MS]). As a consequence, since the value of _{E}µn,n−kf(E) does not depend onF,

we will obtain that, as long as k is in the range considered in the statement of Theorem 1.2, for a set of subspaces with large measure f(E) is very close to its expected value and, therefore, toEµ|x|2for every (n−k)-dimensional faceF. This

will imply that the second term in (3) is bounded by Cnfor every subspaceE in this set. Then, since we have seen before that the first term in (3) is also bounded byCnfor a larger range ofk, we have that

Varµ|x|2≤Cn≤Cλ2µEµ|x|2.

We will denote byO(n) the orthogonal group equipped with the Hilbert-Schmidt distancek · kHS and we represent anyU ∈O(n) by U = (u1, . . . , un), where (ui) is

an orthonormal basis of _{R}n_{. Let us recall that for any linear map}_{T} _{= (t}
ij)ni,j=1,

its Hilbert-Schmidt norm is defined by

kTkHS:=
n
X
i,j=1
t2_{ij}
1
2

and that kTk ≤ kTkHS, where kTk denotes the usual operator norm kTk =

sup_{x}_{∈}_{S}n−1|T x|.

Theorem 3.1 (Concentration of measure). Let f : Gn,n−k → R be a Lipschitz function with Lipschitz constantσ with respect to the distance

d(E1, E2) = inf{kU−VkHS:U, V ∈O(n), E1=span{u1, . . . , un−k},

E2=span{v1, . . . , vn−k}}. Then, for everyλ >0

µn,n−k{E∈Gn,n−k : |f(E)−Ef(E)|> λ|} ≤c1e−
c_{2}λ2n

σ2 _{,}

wherec1 andc2 are positive absolute constants.

In the following lemma we compute the expected value off. Let us point out that what matters to us for our purposes is that, due to the symmetries ofBn

∞, its

value does not depend on the faceF. Nevertheless, we compute its exact value.
Lemma 3.1. Let F be an (n−k)-dimensional face of B_{∞}n. Then

Z

Gn,n−k

EF|PEx|2dµn,n−k(E) =

(n−k)(n+ 2k)

3n .

Proof. Notice that, by Fubini’s theorem and the uniqueness of the Haar measureσ
onSn−1_{we have}
Z
Gn,n−k
EF|PEx|2dµn,n−k(E) = (n−k)
Z
Gn,n−k
EF
Z
SE
hPEx, θi2dσE(θ)dµn,n−k(E)

= (n−k)
Z
Gn,n−k
EF
Z
SE
hx, θi2_{dσ}
E(θ)dµn,n−k(E)
= (n−k)EF
Z
Gn,n−k
Z
SE
hx, θi2dσE(θ)dµn,n−k(E)
= (n−k)EF
Z
Sn−1
hx, θi2_{dσ(θ)}
= n−k
n EF|x|
2_{.}

Using the description of any (n−k) dimensional face F given in (2), a random vector uniformly distributed in the faceF has kcoordinates equal to ±1 and the other n−k coordinates are given by a random vector uniformly distributed on Bn−k

∞ . Thus, the latter expectation equals

n−k
n
k+_{E}_{B}n−k
∞ |x|
2 _{=} n−k
n
k+n−k
3
= n−k
n
n+ 2k
3 .

In the following lemma we estimate the Lipschitz constant off with respect to the distance defined in Theorem 3.1. Notice that, as before, its value does not depend onF.

Lemma 3.2. Let F=F(i1,ε1,...,ik,εk)be an(n−k)-dimensional face ofB

n

∞ and let

f : Gn,n−k →Rbe the function defined as f(E) =EF|PEx|2. For any E1, E2 ∈

Gn,n−k we have

|f(E1)−f(E2)| ≤

8√2k

3 d(E1, E2).

Proof. LetE1, E2∈Gn,n−k. By Lemma 2.2 we have

|f(E1)−f(E2)|=
EF|PE1x|
2_{−}
EF|PE2x|
2
=
PE1
k
X
j=1
εjeij
2
−
PE2
k
X
j=1
εjeij
2
−1
3
k
X
j=1
|PE1(eij)|
2_{− |}_{P}
E2(eij)|
2
≤
PE1
k
X
j=1
εjeij
2
−
PE2
k
X
j=1
εjeij
2
+ 1
3
k
X
j=1
|PE1(eij)|
2_{− |}_{P}
E2(eij)|
2
=
PE1
k
X
j=1
εjeij
+
PE2
k
X
j=1
εjeij
PE1
k
X
j=1
εjeij
−
PE2
k
X
j=1
εjeij
+ 1
3
k
X
j=1
|PE1(eij)|+|PE2(eij)|
|PE1(eij)| − |PE2(eij)|

≤ 2√k (PE1−PE2) k X j=1 εjeij + 2 3 k X j=1 |(PE1−PE2) (eij)| ≤ 2kkPE1−PE2k+ 2k 3 kPE1−PE2k = 8k 3 kPE1−PE2k.

Notice that for any U, V ∈ O(n) such that E1 = span{u1, . . . , un−k} and E2 =

span{v1, . . . , vn−k}, the vectors {uj}nj=1−k and {vj}nj=1−k for orthonormal basis ofE1

andE2 respectively and for anyx∈Rn • PE1x= Pn−k j=1hPE1x, ujiuj = Pn−k j=1hx, ujiuj, • PE2x= Pn−k j=1hPE2x, vjivj= Pn−k j=1hx, vjivj.

Then, for any suchU, V, the projections ontoE1andE2,PE1 andPE2 are given by the matrices (with respect to the canonical basis inRn) (Pnj=1−khuj, ekihuj, eli)

n
k,l=1
and (Pn−k
j=1hvj, ekihvj, eli)nk,l=1 respectively. Thus,
kPE1−PE2k
2 _{≤} _{k}_{P}
E1−PE2k
2
HS = 2(n−k)−2
n−k
X
i,j=1
hui, vji2
≤ 2
n−k
X
j=1
(1− huj, vji2)≤2
n−k
X
j=1
|uj−vj|2≤2
n
X
j=1
|uj−vj|2
= 2kU−Vk2HS,
since 1− huj, vji2 ≤ 2(1− huj, vji) = |uj−vj|2. Consequently kPE1−PE2k ≤
√

2d(E1, E2) and we obtain the result.

Lemma 3.3. Let1≤k≤ n23

(logn)13

. There exist positive absolute constantsC, c1, c2

such that the set E∈Gn,n−k : E F|PEx|2− (n−k)(n+ 2k) 3n > C√n ,for someF

has measure µn,n−k smaller thanc1e−c2n

2

3(logn)23_{.}

Proof. LetF be a fixed (n−k)-dimensional face of Bn

∞. Then, takingλ=C √

n we obtain, using Theorem 3.1 that

µn,n−k
E∈Gn,n−k :
E
F|PEx|2−
(n−k)(n+ 2k)
3n
> C√n
≤c1e−
c_{2}C2n2
k2 _{.}
Since the number of (n−k)-dimensional faces ofBn

∞equals 2k nk ≤2k en k k , using the union bound we have that for anyC >0

µn,n−k
E∈Gn,n−k :
E
F|PEx|2−
(n−k)(n+ 2k)
3n
> C√n ,for someF
≤ c1e−
c_{2}C2n2
k2 +klog 2+klogenk _{≤}_{c}
1e−
c_{2}C2n2
k2 +c3klogn
= c1e−
c2C2n2−c3k3 logn
k2 _{≤}_{c}
1e−(c2C
2_{−}_{c}
3)n
2
3(logn)23
,

Taking into account that c2C2n2−c3k3logn

k2 is decreasing in k and that 1 ≤ k ≤

n23

(logn)13

, the latter term is bounded above by c1e−(c2C

2_{−}_{c}
3)n

2

3(logn)23_{. Choosing} _{C}
a constant big enough we obtain the result. _{}
As a consequence, we obtain the following lemma, which gives an estimate of the
right order for most subspaces, for the second term in (3).

Lemma 3.4. There exists absolute constantsC, c1, c2 such that for any 1 ≤k≤

n23

(logn)13

, the measure µn,n−k of the set of subspaces E∈Gn,n−k for which EPE(F)|x|2−Eµ|x|2 ≤C √ n

for every(n−k)-dimensional faceF ofBn

∞ is greater than1−c1e−c2n

2

3(logn)23_{.}

Proof. Recall that µ denotes the uniform probability measure on K = PE(Bn∞)

and EPE(F) denotes the expectation with respect to the uniform probability on

PE(F). By Lemma 3.3 there exists positive absolute constants C, c1, c2 and a set

of subspaces with measure µn,n−k greater than 1−c1e−c2n

2

3(logn)23 _{such that for}
everyE in such set and every (n−k)-dimensional faceF ofBn

∞,
E
PE(F)|x|
2_{−}(n−k)(n+ 2k)
3n
=
E
F|PEx|2−
(n−k)(n+ 2k)
3n
≤C√n.
Then, for everyF1, F2, (n−k)-dimensional faces

_{E}PE(F1)|x|
2
−EPE(F2)|x|
2
≤2C
√
n.
Consequently, since Eµ|x|2=
l
X
i=1
|PE(Fi)|
|K| EPEFi|x|

2_{, we have that for every} _{E} _{in}

this set and every faceF

EPE(F)|x|
2
−Eµ|x|2≤EPE(F)|x|
2
− min
i=1,...,lEPE(Fi)|x|
2
≤2C√n
and
EF|PEx|2−Eµ|x|2≥EF|PEx|2− max
i=1,...,lEFi|x|
2_{≥ −}_{2C}√_{n.}

Now we are able to prove Theorem 1.2.

Proof of Theorem 1.2. If 1≤k≤ n23

(logn)13

by Lemma 2.2 we have that there exists an absolute constantC such thatλ2µEµ|x|2≥Cn. By equation (3), if{Fi}li=1 are

the (n−k)-dimensional faces ofBn

∞described in (1) we have,
Varµ|x|2≤ max
i=1...,lVarPE(Fi)|x|
2_{+ max}
i=1,...,l EPE(Fi)|x|
2_{−}
Eµ|x|2
2
.

By Lemma 2.5 the first maximum is bounded from above byC1n and by Lemma

3.4 there exists a set of (n−k)-dimensional subspaces with measure larger than 1−c1e−c2n

2 3(logn)23

such that the second maximum is bounded from above byC2n,

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´

Area de an´alisis matem´atico, Departamento de matem´aticas, Facultad de Ciencias, Universidad de Zaragoza, Pedro cerbuna 12, 50009 Zaragoza (Spain), IUMA

E-mail address, (David Alonso): [email protected]

´

Area de an´alisis matem´atico, Departamento de matem´aticas, Facultad de Ciencias, Universidad de Zaragoza, Pedro cerbuna 12, 50009 Zaragoza (Spain), IUMA