Geometry of random sections of isotropic convex bodies

0.28 MB·

by  Apostolos Giannopoulos

Geometry of random sections of isotropic convex
bodies
Apostolos Giannopoulos, Labrini Hioni and Antonis Tsolomitis
6
1
0
2 Abstract

p Let K be an isotropic symmetric convex body in Rn. We show that a subspace F Gn,n−k of
e ∈
codimension k=γn, where γ (1/√n,1), satisfies
S ∈
8 K∩F ⊆ γc√nLK(B2n∩F)
2
withprobabilitygreaterthan1 exp( √n). Usingadifferentmethodwestudythesamequestionforthe
G] Lq-centroidbodiesZq(µ)ofan−isotro−piclog-concaveprobabilitymeasureµon Rn. Forevery16q6n
M Wanedaγls∈o g(i0v,e1)bowuensdhsoowntthhaetdaiarmanetdeormofsruabnsdpoamcepFro∈jecGtnio,(n1s−γo)fnZsqa(tµi)sfiaensdZuqs(iµn)g∩thFem⊆wc2e(dγe)d√uqcBe2nth∩atFif.
K is an isotropic convex body in Rn then for a random subspace F of dimension (logn)4 one has that
.
h all directions in F are sub-Gaussian with constant O(log2n).
t
a
m
1 Introduction
[

2 A convex body K in Rn is called isotropic if it has volume K = 1, its center of mass is at the origin (we
| |
v call these convex bodies “centered”),and its inertia matrix is a multiple of the identity matrix: there exists
4 a constant L >0 such that
K
5
2
2 (1.1) hx,θi2dx=L2K
0 ZK
1. for every θ in the Euclidean unit sphere Sn−1. For every centered convex body K in Rn there exists an
0 invertiblelineartransformationT GL(n)suchthatT(K)isisotropic. ThisisotropicimageofK isuniquely
6 determined up to orthogonal tran∈sformations. A well-known problem in asymptotic convex geometry asks
1
if there exists an absolute constant C >0 such that
: 1
v
i (1.2) Ln :=max LK :K is isotropic in Rn 6C1
X { }
r foralln>1(seeSection2forbackgroundinformationonisotropicconvexbodiesandlog-concaveprobability
a
measures). Bourgainprovedin[5]thatL 6c√4nlogn,andKlartag[19]improvedthis bound toL 6c√4n.
n n
A second proof of Klartag’s bound appears in [21].
Recall that the inradius r(K) of a convex body K in Rn with 0 int(K) is the largest r > 0 for which
∈
rBn K, while the radius R(K):=max x :x K of K is the smallest R>0 for which K RBn. It
is n2ot⊆hard to see that the inradius and t{hke rka2dius∈of a}n isotropic convex body K in Rn satisfy t⊆he bo2unds
c L 6 r(K) 6 R(K) 6 c nL , where c ,c > 0 are absolute constants. In fact, Kannan, Lov´asz and
1 K 2 K 1 2
Simonovits [17] have proved that
(1.3) R(K)6(n+1)L .
K
Radius of random sections of isotropic convex bodies. The first question that we discuss in this
articleis to givesharpupper bounds for the radiusofarandom(n k)-dimensionalsectionofK. Anatural
−
“guess” is that the following question has an affirmative answer.
1
Question 1.1. There exists an absolute constant c > 0 with the following property: for every isotropic
0
convex body K in Rn and for every 16k 6n 1, a random subspace F G satisfies
n,n k
− ∈ −
(1.4) R(K F)6c n/k√nL .
0 K
∩
It was proved in [23] that if K is a symmetric convpex body in Rn then a random F G satisfies
n,n k
∈ −
(1.5) R(K F)6c(n/k)3/2M˜(K),
∩
where c>0 is an absolute constant and
1
(1.6) M˜(K):= x dx.
2
K k k
| |ZK
In the case of an isotropic convex body one has K =1 and
| |
1/2
(1.7) M˜(K)6 x 2dx =√nL ,
k k2 K
(cid:18)ZK (cid:19)
therefore (1.5) implies that a random F G satisfies
n,n k
∈ −
(1.8) R(K F)6c (n/k)3/2√nL ,
1 K
∩
where c >0 is an absolute constant.
1
Our first main result shows that one can have a bound of the order of γ 1√nL when the codimension
− K
k is greater than γn.
Theorem 1.2. Let K be an isotropic symmetric convex body in Rn and let 1 6 k 6 n 1. A random
−
subspace F G satisfies
n,n k
∈ −
c n
(1.9) R(K F)6 0 √nL
K
∩ max k,√n
{ }
with probability greater than 1 exp( √n), where c >0 is an absolute constant.
0
− −
The proof is given in Section 3. Note that Theorem 1.2 gives non-trivial information when k >√n. In
this case, writing k =γn for some γ (1/√n,1) we see that
∈
c
(1.10) R(K F)6 0√nL
K
∩ γ
with probabilitygreaterthan1 exp( √n) onG . The resultof [23]establishesa γ 3/2-dependence
n,(1 γ)n −
− − −
on γ =k/n.
A standard approach to Question 1.1 would have been to combine the low M -estimate with an upper
∗
bound for the mean width
(1.11) w(K):= h (x)dσ(x),
K
ZSn−1
of an isotropic convex body K in Rn, that is, the L -norm of the support function of K with respect to
1
the Haar measure on the sphere. This last problem was open for a number of years. The upper bound
w(K) 6 cn3/4L appeared in the Ph.D. Thesis of Hartzoulaki [16]. Other approaches leading to the same
K
bound can be found in Pivovarov[32] and in Giannopoulos,Paouris and Valettas [15]. Recently, E. Milman
showed in [26] that if K is an isotropic symmetric convex body in Rn then
(1.12) w(K)6c √n(logn)2L .
3 K
2
In fact, it is not hard to see that his argument can be generalized to give the same estimate in the not
necessarily symmetric case. The dependence on n is optimal up to the logarithmic term. From the sharp
versionof V. Milman’s low M -estimate (due to Pajorand Tomczak-Jaegermann[28]; see [2, Chapter 7] for
∗
complete references) one has that, for every 16k 6n 1, a subspace F G satisfies
n,n k
− ∈ −
(1.13) R(K F)6c n/kw(K)
4
∩
p
withprobabilitygreaterthan1 exp( c k),wherec ,c >0areabsoluteconstants. Combining (1.13)with
5 4 5
− −
E. Milman’s theorem we obtain the folowing estimate:
Let K be an isotropic symmetric convex body in Rn. For every 1 6 k 6 n 1, a subspace
−
F G satisfies
n,n k
∈ −
c n(logn)2L
(1.14) R(K F)6 2 K
∩ √k
with probability greater than 1 exp( c k), where c ,c >0 are absolute constants.
3 2 3
− −
NotethattheupperboundofTheorem1.2hassomeadvantageswhencomparedto(1.14): Ifkisproportional
to n (say k > γn for some γ (1/√n,1)) then Theorem 1.2 guarantees that R(K F) 6 c(γ)√nL for a
K
random F G . More ge∈nerally, for all k > c6n we have ∩
∈ n,n−k (logn)4
c n√n c n(logn)2
(1.15) 0 6 2 ,
max k,√n √k
{ }
and hence the estimate of Theorem 1.2 is stronger than (1.14). Nevertheless, we emphasize that our bound
is notoptimal andit wouldbe very interesting to decide whether (1.4) holds true; this wouldbe optimal for
all 16k6n.
Radius of random sections of L -centroid bodies and their polars. In Section 4 we study the
q
diameter ofrandomsections ofthe L -centroidbodies Z (µ) ofanisotropiclog-concaveprobabilitymeasure
q q
µ on Rn. Recall that a measure µ on Rn is called log-concave if µ(λA+(1 λ)B) > µ(A)λµ(B)1 λ for
−
any compact subsets A and B of Rn and any λ (0,1). A function f : Rn −[0, ) is called log-concave
∈ → ∞
if its support f > 0 is a convex set and the restriction of logf on it is concave. It is known that if a
{ }
probability measure µ is log-concaveand µ(H)<1 for every hyperplane H, then µ is absolutely continuous
with respect to the Lebesgue measure and its density f is log-concave; see [4]. Note that if K is a convex
µ
body in Rn then the Brunn-Minkowskiinequality implies that the indicator function 1 of K is the density
K
of a log-concavemeasure.
We say that a log-concave probability measure µ on Rn is isotropic if its barycenter bar(µ) is at the
origin and
x,θ 2dµ(x)=1
ZRnh i
for all θ Sn 1. Note that the normalization is different from the one in (1.1); in particular, a centered
−
convex bo∈dy K of volume 1 in Rn is isotropic if and only if the log-concave probability measure µ with
K
density x Ln1 (x) is isotropic.
7→ K K/LK
The L -centroid bodies Z (µ), q >1, are defined through their support function
q q
1/q
(1.16) h (y):= ,y = x,y qdµ(x) ,
Zq(µ) kh· ikLq(µ) (cid:18)ZRn|h i| (cid:19)
andhaveplayedakeyroleinthestudyofthedistributionoflinearfunctionalswithrespecttothemeasureµ.
For every 16q 6n we obtain sharp upper bounds for the radius of random sections of Z (µ) of dimension
q
proportional to n, thus extending a similar result of Brazitikos and Stavrakakis which was established only
for q [1,√n].
∈
3
Theorem 1.3. Let µ be an isotropic log-concave probability measure on Rn and let 16q 6n. Then:
(i) If k = γn for some γ (0,1), then, with probability greater than 1 e−c4k, a random F Gn,n k
satisfies ∈ − ∈ −
(1.17) R(Z (µ) F)6c (γ)√q,
q 5
∩
where c is an absolute constant and c (γ)=O(γ 2log5/2(c/γ)) is a positive constant depending only
4 5 −
on γ.
(ii) With probability greater than 1 e n, a random U O(n) satisfies
−
− ∈
(1.18) Z (µ) U(Z (µ)) (c √q)Bn,
q ∩ q ⊆ 6 2
where c >0 is an absolute constant.
6
The method of proof is based on estimates (from [26] and [11]) for the Gelfand numbers of symmetric
convex bodies in terms of their volumetric parameters;combining these generalestimates with fundamental
(known)propertiesofthefamilyofthecentroidbodiesZ (µ)ofanisotropiclog-concaveprobabilitymeasure
q
µweprovideestimatesfortheminimalradiusofak-codimensionalsectionofZ (µ). Then,wepasstobounds
q
for the radius of random k-codimensionalsections of Z (µ) using known results from [12], [34] and [24]. We
q
conclude Section 4 with a discussionof the same questions for the polar bodies Z (µ) of the centroidbodies
q◦
Z (µ).
q
Usingthesameapproachwestudythediameterofrandomsectionsofconvexbodieswhichhavemaximal
isotropic constant. Set
(1.19) L :=max L :K is an isotropic symmetric convex body in Rn .
′n { K }
It is known that L 6cL for some absolute constant c>0 (see [9, Chapter 3]). We prove the following:
n ′n
Theorem 1.4. Assume that K is an isotropic symmetric convex body in Rn with L =L . Then:
K ′n
(i) A random F G satisfies
n,n/2
∈
(1.20) R(K F)6c √n
7
∩
and
(1.21) L 6c
K F 8
∩
with probability greater than 1 e−c9n, where ci >0 are absolute constants.
−
(ii) A random U O(n) satisfies
∈
(1.22) K U(K) (c √n)Bn,
∩ ⊆ 10 2
with probability greater than 1 e n, where c >0 is an absolute constant.
− 10
−
The same arguments work if we assume that K has almost maximal isotropic constant, i.e. L > βL
K ′n
for some (absolute) constant β (0,1). We can obtain similar results, with the constants c now depending
i
∈
only on β. It should be noted that Alonso-Guti´errez, Bastero, Bernu´es and Paouris [1] have proved that
every convex body K has a section K F of dimension n k with isotropic constant
∩ −
n en
(1.23) L 6c log .
K F
∩ k k
r
(cid:16) (cid:17)
For the proof of this result they considered an α-regular M-position of K. In Theorem 1.4 we consider
convex bodies in the isotropic position and the estimates (1.20) and (1.21) hold for a random subspace F.
4
Radius of random projections of L -centroid bodies and sub-Gaussian subspaces of isotropic
q
convex bodies. Let K be a centered convex body of volume 1 in Rn. We say that a direction θ Sn 1 is
−
∈
a ψ -direction (where 16α62) for K with constant b>0 if
α
(1.24) ,θ 6b ,θ ,
kh· ikLψα(K) kh· ik2
where
(1.25) ,θ :=inf t>0: exp ( x,θ /t)α dx62 .
kh· ikLψα(K) |h i|
(cid:26) ZK (cid:27)
(cid:0) (cid:1)
Markov’s inequality implies that if K satisfies a ψ -estimate with constant b in the direction of θ then for
α
all t>1 we have x K : x,θ >t ,θ 62e ta/bα. Conversely,one cancheck thattail estimates of
2 −
|{ ∈ |h i| kh· ik }|
this form imply that θ is a ψ -direction for K.
α
It is well-known that every θ Sn 1 is a ψ -direction for K with an absolute constant C. An open
− 1
∈
question is if there exists an absolute constant C > 0 such that every K has at least one sub-Gaussian
direction (ψ -direction) with constant C. It was first proved by Klartag in [20] that for every centered
2
convex body K of volume 1 in Rn there exists θ Sn 1 such that
−
∈
t2
(1.26) x K : x,θ >ct ,θ 2 6e−[log(t+1)]2a
|{ ∈ |h i| kh· ik }|
forallt>1,wherea=3(equivalently, ,θ 6C(logn)a ,θ ). Thisestimatewaslaterimproved
kh· ikLψ2(K) kh· ik2
byGiannopoulos,PaourisandValettasin[14]and[15](seealso[13])whoshowedthatthebodyΨ (K)with
2
support function y ,y has volume
7→kh· ikLψ2(K)
Ψ (K) 1/n
(1.27) c 6 | 2 | 6c logn.
1 2
Z (K)
(cid:18)| 2 |(cid:19)
p
From(1.27)itfollowsthatthereexistsatleastonesub-GaussiandirectionforK withconstantb6C√logn.
Brazitikos and Hioni in [7] proved that if K is isotropic then logarithmic bounds for ,θ hold
kh· ikLψ2(K)
true with probability polynomially close to 1: For any a>1 one has
,θ 6C(logn)3/2max logn,√a L
kh· ikLψ2(K) K
np o
for all θ in a subset Θ of Sn 1 with σ(Θ )>1 n a, where C >0 is an absolute constant.
a − a −
−
Here, we consider the question if one can have an estimate of this type for all directions θ of a subspace
F G of dimensionk increasingto infinity with n. We say thatF G is a sub-Gaussiansubspace for
n,k n,k
∈ ∈
K with constant b>0 if
(1.28) ,θ 6b ,θ
kh· ikLψα(K) kh· ik2
forallθ S :=Sn 1 F. InSection5weshowthatifK isisotropicthenarandomsubspaceofdimension
F −
∈ ∩
(logn)4 is sub-Gaussian with constant b (logn)2. More precisely, we prove the following.
≃
Theorem 1.5. Let K be an isotropic convex body in Rn. If k (logn)4 then there exists a subset Γ of G
n,k
≃
with ν (Γ)>1 n (logn)3 such that
n,k −
−
(1.29) ,θ 6C(logn)2L
kh· ikLψ2(K) K
for all F Γ and all θ S , where C >0 is an absolute constant.
F
∈ ∈
An essential ingredient of the proof is the good estimates on the radius of random projections of the
L -centroidbodiesZ (K)ofK,whichfollowfromE.Milman’ssharpboundsontheirmeanwidthw(Z (K))
q q q
(see Theorem 5.1).
5
2 Notation and preliminaries
We work in Rn, which is equipped with a Euclidean structure , . We denote the correspondingEuclidean
h· ·i
norm by , and write Bn for the Euclidean unit ball, and Sn 1 for the unit sphere. Volume is denoted
k·k2 2 −
by . We write ω for the volume of Bn and σ for the rotationally invariantprobability measure on Sn 1.
|·| n 2 −
We also denote the Haar measure on O(n) by ν. The Grassmann manifold G of k-dimensional subspaces
n,k
of Rn is equipped with the Haar probability measure ν . Let k 6 n and F G . We will denote the
n,k n,k
orthogonalprojection from Rn onto F by P . We also define B =Bn F and∈S =Sn 1 F.
F F 2 ∩ F − ∩
The letters c,c,c ,c etc. denote absolute positive constants whose value may change from line to line.
′ 1 2
Whenever we write a b, we mean that there exist absolute constants c ,c > 0 such that c a 6 b 6 c a.
1 2 1 2
Also ifA,D Rn wew≃ill write A D ifthere existabsolute constantsc ,c >0suchthat c A D c A.
1 2 1 2
⊆ ≃ ⊆ ⊆
Convex bodies. A convex body in Rn is a compact convex subset A of Rn with nonempty interior. We
say that A is symmetric if A = A. We say that A is centered if the center of mass of A is at the origin,
−
i.e. x,θ dx=0 for every θ Sn 1.
Ah i ∈ −
The volume radiusofA is the quantity vrad(A)=(A/Bn )1/n. Integrationinpolarcoordinatesshows
R | | | 2|
that if the origin is an interior point of A then the volume radius of A can be expressed as
1/n
(2.1) vrad(A)= kθk−Andσ(θ) ,
(cid:18)ZSn−1 (cid:19)
where θ = min t > 0 : θ tA . The radial function of A is defined by ρ (θ) = max t > 0 : tθ A ,
A A
k k { ∈ } { ∈ }
θ Sn 1. The support function of A is defined by h (y):=max x,y :x A , and the mean width of A
− A
∈ h i ∈
is the average
(cid:8) (cid:9)
(2.2) w(A):= h (θ)dσ(θ)
A
ZSn−1
ofh onSn 1. The radiusR(A)ofAis the smallestR>0 suchthatA RBn. Fornotationalconvenience
we wArite A f−or the homothetic image of volume 1 of a convex body A ⊆Rn, i.2e. A:= A 1/nA.
−
The polar body A of a convex body A in Rn with 0 int(A) is de⊆fined by | |
◦
∈
(2.3) A := y Rn : x,y 61for allx A .
◦
∈ h i ∈
The Blaschke-Santalo´ inequality states t(cid:8)hat if A is centered then A A (cid:9)6 Bn 2, with equality if and only
| || ◦| | 2|
if A is an ellipsoid. The reverseSantalo´ inequality of J. Bourgainand V. Milman [6] states that there exists
an absolute constant c>0 such that
(2.4) (A A◦ )1/n >c/n
| || |
whenever 0 int(A).
For ever∈y centered convex body A of volume 1 in Rn and for every q ( n, ) 0 we define
∈ − ∞ \{ }
1/q
(2.5) I (A)= x qdx .
q k k2
(cid:18)ZA (cid:19)
As a consequence of Borell’s lemma (see [9, Chapter 1]) one has
(2.6) I (A)6c qI (A)
q 1 2
for all q >2.
For basic facts from the Brunn-Minkowski theory and the asymptotic theory of convex bodies we refer
to the books [33] and [2] respectively.
6
Log-concave probability measures. Let µ be a log-concave probability measure on Rn. The density of
µ is denoted by f . We say that µ is centered and we write bar(µ)=0 if, for all θ Sn 1,
µ −
∈
(2.7) x,θ dµ(x) = x,θ f (x)dx=0.
µ
ZRnh i ZRnh i
The isotropic constant of µ is defined by
1
(2.8) Lµ := supx∈Rnfµ(x) n [detCov(µ)]21n,
f (x)dx
(cid:18) Rn µ (cid:19)
where Cov(µ) is the covariance matrix of µRwith entries
x x f (x)dx x f (x)dx x f (x)dx
(2.9) Cov(µ) := Rn i j µ Rn i µ Rn j µ .
ij
f (x)dx − f (x)dx f (x)dx
R Rn µ R Rn µ R Rn µ
We say that a log-concaveprobabilityRmeasure µ on RnRis isotropic if bRar(µ)=0 and Cov(µ) is the identity
matrix. Note that a centered convex body K of volume 1 in Rn is isotropic,i.e. it satisfies (1.1), if and only
if the log-concave probability measure µ with density x Ln1 (x) is isotropic. Note that for every
log-concavemeasure µ on Rn one has K 7→ K K/LK
(2.10) L 6κL ,
µ n
where κ>0 is an absolute constant (a proof can be found in [9, Proposition 2.5.12]).
We will use the following sharp result on the growth of I (K), where K is an isotropic convex body in
q
Rn, proved by Paouris in [29] and [30].
Theorem 2.1 (Paouris). There exists an absolute constant δ > 0 with the following property: if K is an
isotropic convex body in Rn, then
1 1
(2.11) √nL = I (K)6I (K)6I (K)6δI (K)=δ√nL
K 2 q q 2 K
δ δ −
for every 16q 6√n.
For every q >1 and every y Rn we set
∈
1/q
(2.12) h (y)= x,y qdµ(x) .
Zq(µ) (cid:18)ZRn|h i| (cid:19)
The L -centroid body Z (µ) of µ is the symmetric convex body with support function h . Note that µ
q q Zq(µ)
is isotropic if and only if it is centered and Z (µ) = Bn. If K is an isotropic convex body in Rn we define
2 2
Z (K) = L Z (µ ). From Ho¨lder’s inequality it follows that Z (K) Z (K) Z (K) Z (K) for all
q K q K 1 p q
16p6q 6 , where Z (K)=conv K, K . Using Borell’s lemma,⊆one can ch⊆eck that⊆ ∞
∞ ∞ { − }
q
(2.13) Z (K) c Z (K)
q 1 p
⊆ p
for all 16p <q. In particular, if K is isotropic, then R(Z (K))6c qL . One can also check that if K is
q 1 K
centered, then Z (K) c Z (K) for all q >n.
q 2
It was shown by P⊇aouris∞[29] that if 16q 6√n then
(2.14) w Z (µ) √q,
q
≃
and that for all 16q 6n, (cid:0) (cid:1)
(2.15) vrad(Z (µ))6c √q.
q 1
7
Conversely, it was shown by B. Klartag and E. Milman in [21] that if 16q 6√n then
(2.16) vrad(Z (µ))>c √q.
q 2
This determines the volume radius of Z (µ) for all 1 6 q 6√n. For larger values of q one can still use the
q
lower bound:
(2.17) vrad(Zq(µ))>c2√qL−µ1,
obtained by Lutwak, Yang and Zhang in [25] for convex bodies and extended by Paouris and Pivovarov in
[31] to the class of log-concave probability measures.
Let µ be a probability measure on Rn with density f with respect to the Lebesgue measure. For every
µ
16k 6n 1 and every E G , the marginalof µ with respect to E is the probability measure π (µ) on
n,k E
− ∈
E, with density
(2.18) f (x)= f (y)dy.
πE(µ) µ
Zx+E⊥
It is easily checked that if µ is centered, isotropic or log-concave, then π (µ) is also centered, isotropic or
E
log-concave,respectively. A very useful observation is that:
(2.19) P Z (µ) =Z π (µ)
F q q F
for every 16k6n 1 and every F G (cid:0) . (cid:1) (cid:0) (cid:1)
n,n k
If µ is a centered−log-concavepro∈bability−measure on Rn then for every p>0 we define
(2.20) Kp(µ):=Kp(fµ)= x: ∞rp−1fµ(rx)dr > fµ(0) .
p
(cid:26) Z0 (cid:27)
From the definition it follows that K (µ) is a star body with radial function
p
(2.21) ρKp(µ)(x)= f 1(0) ∞prp−1fµ(rx)dr 1/p
(cid:18) µ Z0 (cid:19)
for x=0. The bodies K (µ) were introduced in [3] by K. Ball who showedthat if µ is log-concavethen, for
p
6
every p>0, K (µ) is a convex body.
p
If K is isotropic then for every 16k 6n 1 and F G , the body K (π (µ )) satisfies
− ∈ n,n−k k+1 F⊥ K
L
(2.22) K F 1/k Kk+1(πF⊥(µK)).
| ∩ | ≃ L
K
For more information on isotropic convex bodies and log-concavemeasures see [9].
3 Random sections of isotropic convex bodies
TheproofofTheorem1.2isbasedonLemma3.1andLemma3.2below. TheyexploitsomeideasofKlartag
from [18].
Lemma 3.1. Let K be an isotropic convex body in Rn. For every 1 6 k 6 n 1 there exists a subset
−
:= (n,k) of G with ν ( )>1 e √n that has the following property: for every F ,
n,n k n,n k −
A A − − A − ∈A
(3.1) x K F : x >c √nL 6e (k+√n) K F ,
2 1 K −
|{ ∈ ∩ k k }| | ∩ |
where c >0 is an absolute constant.
1
8
Proof. Integration in polar coordinates shows that for all q >0
(n k)ω (n k)ω
(3.2) ZGn,n−kZK∩F kxkk2+qdxdνn,n−k(F)= −nωnn−k ZKkxkq2dx= −nωnn−kIqq(K),
and an application of Markov’s inequality shows that a random F G satisfies
n,n k
∈ −
(n k)ω
(3.3) kxkk2+qdx6 −nω n−k(eIq(K))q
ZK∩F n
with probability greater than 1 e q.
−
−
Fix a subspace F G which satisfies (3.3). From (2.22) we have
n,n k
∈ −
(3.4) K F 1/k >c2LKk+1(πF⊥(µK)) > c3
| ∩ | L L
K K
where c ,c >0 are absolute constants. A simple computation shows that
2 3
(n k)ω
(3.5) − n−k 6(c4√n)k
nω
n
for an absolute constant c >0. Using also (2.11) with q =√n we get
4
1 1 (n k)ω
(3.6) K F kxkk2+√ndx6 K F −nω n−k(eI√n(K))√n
| ∩ |ZK∩F | ∩ | n
6(c L )k(c √n)k(eδ√nL )√n 6(c √nL )k+√n,
5 K 4 K 6 K
where c >0 is an absolute constant. It follows that
6
(3.7) x K F : x >ec √nL 6e (k+√n) K F .
2 6 K −
|{ ∈ ∩ k k }| | ∩ |
and the lemma is proved with c =ec . ✷
1 6
The next lemma comes from [18].
Lemma 3.2 (Klartag). Let A be a symmetric convex body in Rm. Then, for any 0<ε<1 we have
1
(3.8) x A: x >εR(A) > (1 ε)m A.
2
|{ ∈ k k }| 2 − | |
Proof. Let x A such that x =R(A) and define v =x / x . We consider the set A+ defined as
0 0 2 0 0 2
∈ k k k k
(3.9) A+ := x A: x,v >0 .
{ ∈ h i }
Since A is symmetric, we have A+ = A/2. Note that
| | | |
(3.10) x A: x >εR(A) εx +(1 ε)A+.
2 0
{ ∈ k k }⊇ −
Therefore,
1
(3.11) x A: x >εR(A) > εx +(1 ε)A+ =(1 ε)m A+ = (1 ε)m A,
2 0
|{ ∈ k k }| | − | − | | 2 − | |
as claimed. ✷
Proof of Theorem 1.2. Let K be an isotropic symmetric convex body in Rn. Applying Lemma 3.1 we
find a subset of G with ν ( )>1 e √n such that, for every F ,
n,n k n,n k −
A − − A − ∈A
(3.12) x K F : x >c √nL 6e (k+√n) K F .
2 1 K −
|{ ∈ ∩ k k }| | ∩ |
9
We distinguish two cases:
Case 1. If k >n/3 then choosing ε0 =1 e−31 we get
−
1 1
(3.13) (1 ε0)n−k K F = e−n−3k K F >e−n−3k−1 K F >e−(k+√n) K F ,
2 − | ∩ | 2 | ∩ | | ∩ | | ∩ |
because k+√n> n k +1. By Lemma 3.2 and (3.12) we get that
−3
(3.14) x K F : x >ε R(K F) > x K F : x >c √nL ,
2 0 2 1 K
|{ ∈ ∩ k k ∩ }| |{ ∈ ∩ k k }|
therefore
(3.15) R(K F)<c √nL ,
2 K
∩
where c2 =ε−01c1 >0 is an absolute constant.
Case 2. If k 6 n/3 then we choose ε = k+√n. Note that ε < 1/2. Using the inequality 1 t > e 2t on
1 6(n k) 1 − −
−
(0,1/2) we get
n k
(3.16) 1(1 ε1)n−k K F = 1 1 k+√n − K F >e−k+3√n−1 K F >e−(k+√n) K F ,
2 − | ∩ | 2 − 6(n k) | ∩ | | ∩ | | ∩ |
(cid:18) − (cid:19)
because 2(k+√n) >1. By Lemma 3.2 this implies that
3
(3.17) x K F : x >ε R(K F) > x K F : x >c √nL ,
2 1 2 1 K
|{ ∈ ∩ k k ∩ }| |{ ∈ ∩ k k }|
therefore
(3.18) ε R(K F)<c √nL ,
1 1 K
∩
which, by the choice of ε becomes
1
c n
(3.19) R(K F)< 3 √nL
K
∩ max k,√n
{ }
for some absolute constant c > 0. This completes the proof of the theorem (with a probability estimate
3
1 e √n for all 16k6n 1). ✷
−
− −
Remark 3.3. It is possible to improve the probability estimate 1 e √n in the range k > γn, for any
−
−
γ (1/√n,1). Thiscanbe donewiththe helpofknownresultsthatdemonstratethe factthattheexistence
∈
of one s-dimensional section with radius r implies that random m-dimensional sections, where m<s, have
radius of “the same order”. This was first observed in [12], [34] and, soon after, in [24]. Let us recall this
last statement.
Let A be a symmetric convex body in Rn and let 16s<m6n 1. If R(A F)6r for some
− ∩
F G then a random subspace E G satisfies
n,m n,s
∈ ∈
(3.20) R(A E)6r c2n 2(nm−−ss)
∩ n m
(cid:16) − (cid:17)
with probability greater than 1 2e (n s)/2, where c >0 is an absolute constant.
− − 2
−
We applythis resultasfollows. Letk =γn>√nandsett=δn,whereδ γ/log(1+1/γ). Fromthe proof
≃
of Theorem 1.2 we know that there exists E G such that
n,n t
∈ −
c n
(3.21) R(K E)6 1 √nL ,
K
∩ t
10