Asymptotic shape of the convex hull of isotropic log-concave random vectors

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by  Apostolos Giannopoulos

Asymptotic shape of the convex hull of isotropic
log-concave random vectors
Apostolos Giannopoulos, Labrini Hioni and Antonis Tsolomitis
6
1
0
2 Abstract

n Let x1,...,xN be independent random points distributed according to an isotropic log-concave
a measure µon Rn,and consider therandom polytope
J

0 KN :=conv x1,..., xN .
{± ± }
1
We provide sharp estimates for the quermaßintegrals and other geometric parameters of KN in the

] range cn 6N 6exp(n); these complement previous results from [13] and [14] that were given for the
G
rangecn6N 6exp(√n). Oneofthebasicnewingredientsinourwork isarecentresult ofE.Milman
M that determines themean width of thecentroid body Zq(µ) of µfor all 16q6n.
.
h
1 Introduction
t
a
m
The purpose of this work is to add new information on the asymptotic shape of random polytopes whose
[
vertices have a log-concavedistribution. Without loss of generality we shall assume that this distribution is

1 also isotropic. Recall that a convex body K in Rn is calledisotropic if it has volume 1, it is centered, i.e. its
v center of mass is at the origin, and its inertia matrix is a multiple of the identity: there exists a constant
2 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. More generally, a log-concave probability measure µ on Rn
−
0 is called isotropic if its center of mass is at the origin and its inertia matrix is the identity; in this case, the
6
isotropic constant of µ is defined as
1
:
v (1.2) L := sup f (x) 1/n,
µ µ
Xi x∈Rn
(cid:0) (cid:1)
r where fµ is the density of µ with respect to the Lebesgue measure. Note that a centered convex body
a K of volume 1 in Rn is isotropic if and only if the log-concave probability measure µ with density x
K
7→
Ln1 (x) is isotropic.
K K/LK
Averywell-knownopenquestioninthe theoryofisotropicmeasuresisthe hyperplaneconjecture,which
asks if there exists an absolute constant C >0 such that
(1.3) L :=sup L :µ is an isotropic log-concave measure on Rn 6C
n µ
{ }
for all n > 1. Bourgain proved in [9] that L 6 c√4nlogn (more precisely, he showed that L 6 c√4nlogn
n K
for every isotropic symmetric convex body K in Rn), while Klartag [18] obtained the bound L 6 c√4n. A
n
second proof of Klartag’s estimate appears in [20].
The study of the asymptotic shape of random polytopes whose vertices have a log-concave distribution
was initiated in [13] and [14]. Given an isotropic log-concave measure µ on Rn, for every N > n we
considerN independentrandompointsx ,...,x distributedaccordingtoµanddefinetherandompolytope
1 N
1
K :=conv x ,..., x . The main idea in these works was to compare K with the L -centroid body
N 1 N N q
{± ± }
of µ for a suitable value of q; roughlyspeaking, K is close to the body Z (µ) with high probability.
N log(2N/n)
Recall that the L -centroid bodies Z (µ), q >1, are defined through their support function h , which is
q q Zq(µ)
given by
1/q
(1.4) h (y):= ,y = x,y qdµ(x) .
Zq(µ) kh· ikLq(µ) (cid:18)ZRn|h i| (cid:19)
These bodies incorporate information about the distribution of linear functionals with respect to µ. The
L -centroid bodies were introduced, under a different normalization, by Lutwak and Zhang in [23], while in
q
[29] for the first time, and in [30] later on, Paouris used geometric properties of them to acquire detailed
information about the distribution of the Euclidean norm with respect to µ.
It was proved in [13] that, given any isotropic log-concave measure µ on Rn and any cn 6 N 6 en, the
random polytope K defined by N independent random points x ,...,x which are distributed according
N 1 N
to µ satisfies, with high probability, the inclusion
(1.5) K c Z (µ)
N 1 log(N/n)
⊇
(for the precise statement see Fact 3.2). Then, using the fact that the volume of the L -centroid bodies
q
satisfies the lower bounds Z (µ)1/n >c q/n if q 6√n and Z (µ)1/n >c L 1 q/n if √n6q 6n (see
| q | 2 | q | 3 −µ
Section 2), we see that for n6N 6e√n we have
p p
log(2N/n)
(1.6) K 1/n >c ,
N 4
| | √n
p
while in the range e√n 6N 6en we have
log(2N/n)
(1.7) K 1/n >c L 1
| N| 5 −µ √n
p
with probability exponentially close to 1. On the other hand, one can check that for every α>1 and q >1,
(1.8) E σ ( θ :h (θ)>αh (θ) ) 6Nα q,
n { KN Zq(µ) } −
whereσ isthe rotationallyinvari(cid:2)antprobabilitymeasureonthe E(cid:3)uclideanunitsphereSn 1. Thisestimate
n −
is sufficient for some sharp upper bounds. First, for all n6N 6exp(n) one has
(1.9) E w(K ) 6c w(Z (µ)),
N 6 logN
where the mean width w(C) of a convex(cid:2)body C (cid:3)in Rn containing the origin,is defined as twice the average
of its support function on Sn 1:
−
w(C)= h (θ)dσ (θ).
C n
ZSn−1
Second, one has
log(2N/n)
(1.10) K 1/n 6c
N 7
| | √n
p
with probability greater than 1 1, where C >0 is an absolute constant.
− N
In [14] these results were extended to the full family of quermaßintegrals W (K ) of K . These are
n k N N
−
defined through Steiner’s formula
n
n
(1.11) K+tBn = W (K)tn k,
| 2| k n−k −
k=0(cid:18) (cid:19)
X
2
whereW (K)isthemixedvolumeV(K,k;Bn,n k). Itismoreconvenienttoexpresstheestimatesusing
n−k 2 −
a normalized variant of W (K): for every 16k 6n we set
n k
−
1/k 1/k
W (K) 1
n k
(1.12) Qk(K)= − = PF(K) dνn,k(F) ,
(cid:18) ωn (cid:19) ωk ZGn,k| | !
where the last equality followsfrom Kubota’s integralformula (see Section 2 for backgroundinformationon
mixed volumes). Then, one has the following results on the expectation of Q (K ) for all values of k:
k N
Theorem 1.1 (Dafnis, Giannopoulos and Tsolomitis, [14]). If n2 6N 6exp(cn) then for every 16k 6n
we have
(1.13) L 1 logN .E Q (K ) .w(Z (K)).
−µ k N logN
In the range n2 6N 6exp(√n) onephas an asym(cid:2)ptotic fo(cid:3)rmula: for every 16k 6n,
(1.14) E Q (K ) logN.
k N
≃
All these estimates remain valid for n1+(cid:2)δ 6 N 6(cid:3)n2,pwhere δ (0,1) is fixed, if we allow the constants
∈
to depend onδ. Working in the range N n is possible, but requires some additional attention (see e.g. [5]
≃
for the case of mean width).
A more careful analysis (which can be found in [14, Theorem 1.2]) shows that if n2 6 N 6 exp(√n)
then, for any s>1, a random K satisfies, with probability greater than 1 N s,
N −
−
(1.15) Q (K )6c (s) logN
k N 1
for all 16k6n and, with probability greater than 1 expp( √n),
− −
(1.16) Q (K )>c logN
k N 8
for all 16k6n, where c (s)>0 depends only on s, andpc >0 is an absolute constant.
1 8
Anaturalquestionthatarisesiswhethertheseresultscanbeextendedtothefullrangecn6N 6exp(n)
of values of N. If one decides to follow the approach of [13] and [14] then there are two main obstacles.
The first one is that the lower bound Z (µ)1/n >c q/n is currently known only in the range q 6√n. In
q
| |
fact, proving the same for larger values of q would lead to improved estimates on L (for example, see the
p n
computationafterLemma2.2in[20]). Thesecondonewasthat,untilrecently,asharpestimateonthemean
width of Z (µ) was known only for q 6 √n; G. Paouris proved in [29] that for every isotropic log-concave
q
measure µ on Rn and any q 6√n one has
(1.17) w Z (K) 6c √q.
q 9
Recently, E. Milman [25] obtained the same u(cid:0)pper bo(cid:1)und (modulo logarithmic terms) for q beyond √n.
Theorem 1.2 (E. Milman, [25]). For every isotropic log-concave measure µ on Rn and for all q [√n,n]
∈
we have
(1.18) w(Z (µ))6c √q log2(1+q).
q 10
Animmediate consequenceofthis resultis thatitprovidesanew boundforthe meanwidthofanorigin
symmetric isotropic convexbody K in Rn. In this caseit is knownthat Z (K) cK, andwe conclude that
n
⊇
(1.19) w(K)6C √n log2(1+n)L
1 K
improving the earlier known bound w(K) 6 C n3/4L of Hartzoulaki, from her PhD thesis [17]. We note
2 K
here that not all of the logarithmic terms in (1.19) can be removed, as the example of Bn/Bn 1/n shows.
1 | 1|
Using E. Milman’s theorem we can show the following.
3
Theorem1.3. Letx ,...,x beindependentrandompointsdistributedaccordingtoanisotropiclog-concave
1 N
measure µ on Rn, and consider the random polytope K := conv x ,..., x . If exp(√n) 6 N 6
N 1 N
{± ± }
exp(cn) then for every 16k 6n we have
(1.20) L 1 logN .E Q (K ) . logN loglogN 2.
−µ k N
p (cid:2) (cid:3) p (cid:0) (cid:1)
Next we provide estimates for Q (K ) for “most” K :
k N N
Theorem1.4. Letx ,...,x beindependentrandompointsdistributedaccordingtoanisotropiclog-concave
1 N
measure µ on Rn, and consider the random polytope K :=conv x ,..., x . For all exp(√n)6 N 6
N 1 N
{± ± }
exp(n) and s>1 we have
(1.21) Q (K )6c (s) logN (loglogN)2,
k N 2
for all 16k <n, with probability greater than 1 Np s.
−
−
WealsoprovideestimatesonthevolumeradiusofarandomprojectionP (K )ofK ontoF G (in
F N N n,k
∈
termsofn,k andN)inthe rangee√n 6N 6en; these extendthe sharpestimate v.rad(P (K )) √logN
F N
≃
that was obtained in [14] for the case N 6e√n.
Theorem 1.5. If exp(√n) 6 N 6 ecn and s > 1, then a random K satisfies with probability greater
N
than 1 max N s,e c11√N the following: for every 1 6 k 6 n there exists a subset M of G with
− − n,k n,k
− { }
ν (M )>1 e c12k such that
n,k n,k −
−
1/k
P (K )
c13L−µ1 logN 6v.rad(PF(KN)):= | Fω N |
(cid:18) k (cid:19)
p 2
(1.22) 6c (s) logN loglogN
3
for all F M . p (cid:0) (cid:1)
n,k
∈
InSection4weprovideanalternativeproofofanestimateofAlonso-Guti´errez,Dafnis,Hern´andez-Cifre
and Prochno from [3] on the k-th mean outer radius
(1.23) R˜ (K )= R(P (K ))dν (F)
k N F N n,k
ZGn,k
of a random K , as a function of N,n and k.
N
Theorem1.6. Letx ,...,x beindependentrandompointsdistributedaccordingtoanisotropiclog-concave
1 N
measure µ on Rn, and consider the random polytope K := conv x ,..., x . If n 6 N 6 exp(√n)
N 1 N
{± ± }
then, for all 16k 6n and s>0 one has
(1.24) c (s)max √k, log(N/n) 6R˜ (K )6c (s)max √k, logN
4 k N 5
n p o n p o
with probability greater than 1 N s, where c (s),c (s) are positive constants depending only on s.
− 4 5
−
We provideaformulaforR˜ (K )whichisvalidforallcn6N 6exp(n). This allowsustorecover(and
k N
explain) the sharp estimate of Theorem 1.6 for “small” values of N and to obtain its analogue for “large”
values of N; see Theorem 4.5.
InSection5weobtainestimatesontheregularityofthecoveringnumbersandthedualcoveringnumbers
of a random K . In a certain range of values of N, these allow us to conclude that a random K is in
N N
α-regular M-position with α 1 (see Section 5 for definitions and terminology).
∼
4
Theorem 1.7. Let µ be an isotropic log-concave measure on Rn. Then, assuming that n2 6 N 6
exp (nlogn)2/5 , we have that a random K satisfies with probability greater than 1 N 1 the entropy
N −
−
estimates
(cid:0) (cid:1) n(logn)2log(1+t)
max logN(K ,tr Bn),logN(r Bn,tK ) 6c
{ N N 2 N 2 N } 14 t
for every t>1, where r =√logN and c >0 is an absolute constant.
N 14
As an application we estimate the average diameter of k-dimensional sections of a random K , defined
N
by
(1.25) D˜ (K )= R(K F)dν (F).
k N N n,k
∩
ZGn,k
The discussion shows that the behavior of D˜ (K ) is not always the same as that of R˜ (K ). In order to
k N k N
give an idea of the results, let us mention here the following simplified version.
Theorem 1.8. Let µ be an isotropic log-concave measure on Rn and a,b (0,1).
∈
(i) If k 6bn then a random K satisfies with probability 1 N 1
N −
−
D˜ (K )6c logN if n2 6N 6exp(√n)
k N b
and p
D˜ (K )6c logN(loglogN)2 if exp(√n)6N 6exp(n).
k N b
(ii) If k >an and N 6exp((nlogn)2p/5) then a random K satisfies with probability 1 exp( √n)
N
− −
√logN
c 6D˜ (K ).
a log3n k N
where c ,c are positive constants that depend only on a and b respectively.
a b
We conclude this paper with a brief discussion of the interesting (open) question whether the isotropic
constant of a random K is bounded by a constant independent from n and N. The first class of random
N
polytopesK inRn forwhichuniformboundswereestablishedwastheclassofGaussianrandompolytopes.
N
Klartag and Kozma proved in [19] that if N > n and if G ,...,G are independent standard Gaussian
1 N
random vectors in Rn, then the isotropic constant of the random polytope K = conv G ,..., G is
N 1 N
{± ± }
bounded by an absolute constant C > 0 with probability greater than 1 Ce cn. The same idea works in
−
−
thecasewheretheverticesx ofK aredistributedaccordingtoanisotropicψ -measureµ; theboundthen
j N 2
depends onlyonthe ψ -constantofµ. Alonso-Guti´errez[2]andDafnis,GiannopoulosandGu´edon[12]have
2
applied the same more or less method to obtain a positive answer in the case where the vertices of K are
N
chosen from the unit sphere or an unconditional isotropic convex body respectively. We show that, in the
general isotropic log-concave case, the method of Klartag and Kozma gives the bound O( log(2N/n)) if
N 6exp(√n) (a proofalong the same lines and an extension to randomperturbations of randompolytopes
p
appear in [4]).
2 Notation and background material
We work in Rn, which is equipped with a Euclidean structure , . We denote by the corresponding
2
h· ·i k·k
Euclideannorm, andwrite Bn for the Euclideanunit ball, and Sn 1 for the unit sphere. Volume is denoted
2 −
by . Wewriteω forthe volumeofBn andσ forthe rotationallyinvariantprobabilitymeasureonSn 1.
The|·G| rassmann mnanifold G of k-di2mensionnal subspaces of Rn is equipped with the Haar probabil−ity
n,k
measure ν . Let 1 6 k 6 n and F G . We will denote the orthogonal projection from Rn onto F by
n,k n,k
∈
P . We also define B =Bn F and S =Sn 1 F.
F F 2 ∩ F − ∩
5
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
Similarly,if K,L Rn≃we will write K L if there exist absoluteconstants c ,c >0 suchthat c K L
1 2 1
c K. We also wri⊆te A for the homothet≃ic image of volume 1 of a convex body A Rn, i.e. A:= A⊆. ⊆
2 ⊆ A1/n
AconvexbodyisacompactconvexsubsetC ofRn withnon-emptyinterior. WesaythatC iss|y|mmetric
if x C wheneverx C. WesaythatC iscenteredifithascenterofmassattheorigini.e. x,θ dx=0
for−ev∈ery θ Sn 1. T∈he support function h : Rn R of C is defined by h (x) = max xC,yh : yi C .
− C C
∈ → {hR i ∈ }
For each <p< , p=0, we define the p-mean width of C by
−∞ ∞ 6
1/p
(2.1) w (C):= hp(θ)dσ (θ) .
p C n
(cid:18)ZSn−1 (cid:19)
ThemeanwidthofCisthequantityw(C)=w (C). TheradiusofCisdefinedasR(C)=max x :x C
1 2
{k k ∈ }
and, if the origin is an interior point of C, the polar body C of C is
◦
(2.2) C := y Rn : x,y 61for allx C .
◦
{ ∈ h i ∈ }
Finally, if C is a symmetric convex body in Rn and is the norm induced to Rn by C, we set
C
k·k
M(C)= x dσ (x)
C n
ZSn−1k k
and write b(C) for the smallestpositive constant b with the property x 6b x for all x Rn. From V.
C 2
k k k k ∈
Milman’s proof of Dvoretzky’s theorem (see [6, Chapter 5]) we know that if k 6 cn(M(C)/b(C))2 then for
most F G we have C F 1 B .
∈ n,k ∩ ≃ M(C) F
2.1 Quermaßintegrals
Let denote the class of non-empty compact convex subsets of Rn. The relation between volume and
n
K
the operations of addition and multiplication of compact convex sets by nonnegative reals is described by
Minkowski’s fundamental theorem: If K ,...,K , m N, then the volume of t K + +t K is a
1 m n 1 1 m m
∈K ∈ ···
homogeneous polynomial of degree n in t >0:
i
(2.3) t K + +t K = V(K ,...,K )t t ,
| 1 1 ··· m m| i1 in i1··· in
16i1,X...,in6m
wherethecoefficientsV(K ,...,K )canbe chosentobe invariantunderpermutationsoftheirarguments.
i1 in
The coefficient V(K ,...,K ) is called the mixed volume of the n-tuple (K ,...,K ).
i1 in i1 in
Steiner’s formula is a special case of Minkowski’s theorem; if K is a convex body in Rn then the volume
of K+tBn, t>0, can be expanded as a polynomial in t:
2
n
n
(2.4) K+tBn = W (K)tn k,
| 2| k n−k −
k=0(cid:18) (cid:19)
X
where W (K) := V(K,k;Bn,n k) is the (n k)-th quermaßintegral of K. It will be convenient for us
n−k 2 − −
to work with a normalized variant of W (K): for every 16k 6n we set
n k
−
1/k
1
(2.5) Q (K)= P (K) dν (F) .
k F n,k
ωk ZGn,k| | !
Note that Q (K)=w(K). Kubota’s integral formula
1
ω
n
(2.6) W (K)= P (K)dν (F)
n k F n,k
− ωk ZGn,k| |
6
shows that
W (K) 1/k
n k
(2.7) Qk(K)= − .
ω
(cid:18) n (cid:19)
The Aleksandrov-Fenchelinequality states that if K, L, K ,...,K , then
3 n n
∈K
(2.8) V(K,L,K ,...,K )2 >V(K,K,K ,...,K )V(L,L,K ,...,K ).
3 n 3 n 3 n
This implies that the sequence (W (K),...,W (K)) is log-concave: we have
0 n
(2.9) Wjk−i >Wik−jWkj−i
if 0 6 i < j < k 6 n. Taking into account (2.7) we conclude that Q (K) is a decreasing function of k. For
k
the theory of mixed volumes we refer to [33].
2.2 L -centroid bodies of isotropic log-concave measures
q
We denote by the class of all Borel probability measures on Rn which are absolutely continuous with
n
P
respect to the Lebesgue measure. The density of µ is denoted by f . We say that µ is centered
n µ n
∈P ∈P
if, for all θ Sn 1,
−
∈
(2.10) x,θ dµ(x) = x,θ f (x)dx=0.
µ
ZRnh i ZRnh i
A measure µ on Rn is called log-concave if µ(λA+(1 λ)B) > µ(A)λµ(B)1 λ for all compact subsets A
−
and B of Rn and all 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 to it is concave. Borell has provedin [8] that if a probability measure
µ is log-concaveand µ(H)<1 for every hyperplane H, then µ and its density f is log-concave. Note
n µ
that if K is a convex body of volume 1 in Rn then the Brunn-∈MPinkowski inequality implies that 1 is the
K
density of a log-concave measure.
If µ is a log-concave measure on Rn with density f , we define the isotropic constant of µ by
µ
1
(2.11) 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
(2.12) 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
Note that L is an affine invariant ofRµ and does not deRpend on the chRoice of the Euclidean structure. We
µ
say that a log-concave probability measure µ on Rn is isotropic if it is centered and Cov(µ) is the identity
matrix.
Recallthatifµisalog-concaveprobabilitymeasureonRn andifq >1thenthe L -centroidbodyZ (µ)
q q
of µ is the symmetric convex body with support function
1/q
(2.13) h (y):= x,y qdµ(x) .
Zq(µ) (cid:18)ZRn|h i| (cid:19)
Observe that µ is isotropic if and only if it is centered and Z (µ)=Bn. From Ho¨lder’s inequality it follows
2 2
that Z (µ) Z (µ) Z (µ) for all 1 6 p 6 q < . Conversely, using Borell’s lemma (see [28, Appendix
1 p q
⊆ ⊆ ∞
III]), one can check that
q
(2.14) Z (µ) c Z (µ)
q 1 p
⊆ p
7
for all 16p<q. In particular, if µ is isotropic, then R(Z (µ))6c q.
q 2
For any α>1 and any θ Sn 1 we define the ψ -norm of x x,θ as follows:
− α
∈ 7→h i
x,θ α
(2.15) ,θ :=inf t>0: exp |h i| dµ(x)62 ,
kh· ikψα (cid:26) ZRn (cid:18)(cid:16) t (cid:17) (cid:19) (cid:27)
provided that the set on the right hand side is non-empty. We say that µ satisfies a ψ -estimate with
α
constant b =b (θ) in the direction of θ if we have
α α
,θ 6b ,θ .
kh· ikψα αkh· ik2
We say that µ is a ψ -measure with constant B >0 if
α α
,θ
sup kh· ikψα 6B .
α
θ∈Sn−1 kh·,θik2
From Borell’slemma it follows that everylog-concavemeasure is a ψ -measurewith constantC, where C is
1
an absolute positive constant.
From [29] and [30] one knows that the “q-moments”
1/q
(2.16) I (µ):= x qdx , q ( n,+ ) 0 ,
q (cid:18)ZRnk k2 (cid:19) ∈ − ∞ \{ }
of the Euclidean norm with respect to an isotropic log-concave probability measure µ on Rn are equivalent
toI (µ)=√naslongas q 6√n. Twomainconsequencesofthis factare: (i)Paouris’deviationinequality
2
| |
(2.17) µ( x Rn : x >c t√n )6exp t√n
2 3
{ ∈ k k } −
for every t > 1, where c > 0 is an absolute constant, and (ii) Paou(cid:0)ris’ sm(cid:1)all ball probability estimate: for
3
any 0<ε<ε , one has
0
(2.18) µ( x Rn : x <ε√n )6εc4√n,
2
{ ∈ k k }
where ε ,c >0 are absolute constants.
0 4
The next theorem summarizes our knowledge on the mean width of Z (µ). The first statement was
q
proved by Paouris in [29], while the second one is E. Milman’s Theorem 1.2.
Theorem 2.1. Let µ be an isotropic log-concave measure on Rn. If 16q 6√n, then
(2.19) w(Z (µ)) √q.
q
≃
Moreover, for all q [√n,n] we have
∈
(2.20) w(Z (µ))6c √q log2(1+q).
q 5
Thenexttheoremsummarizesourknowledgeonthe volumeradiusofZ (µ). Thefirststatementfollows
q
fromthe resultsof[29]and[20],while thelefthand-sideinthe secondonewasobtainedin[24]andthe right
hand-side in [29].
Theorem 2.2. Let µ be an isotropic log-concave measure on Rn. If 16q 6√n then
(2.21) Z (µ)1/n q/n,
q
| | ≃
while if √n6q 6n then p
(2.22) c L 1 q/n6 Z (µ)1/n 6c q/n.
6 −µ | q | 7
The reader may find a detailed expospition of the theory of ispotropic log-concave measures in the book
[11].
8
3 Estimates for the Quermaßintegrals
We start with the proof of Theorem 1.3. Recall that the equivalence E Q (K ) √logN in the range
k N
≃
n2 6 N 6 exp(√n) was proved in [14] (see Theorem 1.1). What is new is the right hand-side estimate in
(1.20). However,in[14] itwasprovedthatE[Q (K )]6w(Z (K))for(cid:2)the full r(cid:3)angeofN. So the result
k N logN
follows immediately by applying Theorem 2.1.
To proveTheorem1.4 we will need Lemma 4.2 from[14] which holds true in the more generalsetting of
isotropic log-concave random vectors.
Lemma 3.1. Let µ be an isotropic log-concave measure on Rn. For every n2 6N 6exp(cn) and for every
q >logN and r>1, we have
hq (θ)
(3.1) KN dσ (θ)6(c r)q
ZSn−1 hqZq(µ)(θ) n 1
with probability greater than 1 r q, where c >0 is an absolute constant.
− 1
−
Proof of Theorem 1.4. Let exp(√n)6N 6exp(n). Applying Ho¨lder’s inequality we get
w(K )= h (θ)dσ (θ)
N KN n
ZSn−1
1/p q 1/q
6 h (θ) pdσ (θ) hKN(θ) dσ (θ)
(cid:18)ZSn−1 Zq(µ) n (cid:19) (cid:18)ZSn−1(cid:18)hZq(µ)(θ)(cid:19) n (cid:19)
(cid:0) (cid:1)
q 1/q
h (θ)
=w Z (µ) KN dσ (θ) ,
p q n
(cid:18)ZSn−1(cid:18)hZq(µ)(θ)(cid:19) (cid:19)
(cid:0) (cid:1)
where p is the conjugate exponent of q. If we now choose q =logN >√n and use Lemma 3.1 we arrive at
w(K )6c rw Z (µ)
N 1 p q
withprobabilitygreaterthan1 r q. Since q =logN itfo(cid:0)llowst(cid:1)hatp<2andthus w (Z (µ))isequivalent
− p q
−
to w(Z (µ)) (see [6, Chapter 5]). Using this and applying Theorem 1.2 we conclude that
q
2
w(K )6c r logN loglogN
N 2
with probability greater than 1 r logN. Choosinpg r =e(cid:0)we compl(cid:1)ete the proof of (1.21). ✷
−
−
We can also give estimates on the volume radius of a random projection P (K ) of K onto F G
F N N n,k
∈
in terms of n,k and N. In [14] it was shown that if n2 6 N 6 exp(√n) then, a random K satisfies with
N
probability greater than 1 N s the following: for every 16k 6n,
−
−
(3.2) c logN 6v.rad(P (K ))6c (s) logN
3 F N 4
withprobabilitygreaterthan1 epc5k withrespecttotheHaarmeaspureν onG . Weextendthisresult
− n,k n,k
−
to the case exp(√n)6N 6exp(n).
For the proof we will use Theorem 1.1 from [13], which was already mentioned in the introduction. We
formulate it in the more general setting of isotropic log-concave random vectors (the probability estimate
in the statement makes use of [1, Theorem 3.13]: if γ > 1 and Γ : ℓn ℓN is the random operator
Γ(y) = ( x ,y ,... x ,y ) defined by the vertices x ,...,x of K the2n→P( Γ2 : ℓn ℓN > γ√N) 6
h 1 i h N i 1 N N k 2 → 2 k
exp( c γ√N) for all N >cγn—see [13] for the details).
0
−
9
Fact 3.2. Letµ bean isotropic log-concavemeasureonRn andletx ,...,x beindependent random vectors
1 N
distributed according to µ, with N >c n where c >1 is an absolute constant. Then, for all q 6c log(N/n)
1 1 2
we have that
(3.3) K c Z (µ)
N 3 q
⊇
with probability greater than 1 exp( c √N).
4
− −
Proof of Theorem 1.5. For the upper bound we use (1.21) and Kubota’s formula to get
1/k
1 2
P (K ) dν (F) 6c (s) logN loglogN L .
F N n,k 6 K
ωk ZGn,k| | !
p (cid:0) (cid:1)
Applying now Markov’s inequality we get that with probability greater than 1 t k with respect to the
−
−
Haar measure ν on G we have
n,k n,k
1/k
|PF(KN)| 6c (s)t logN loglogN 2.
6
ω
(cid:18) k (cid:19)
p (cid:0) (cid:1)
Choosing t=e proves the result.
For the lower bound integrating in polar coordinates and using Ho¨lder’s inequality we have
P (K ) 1
(3.4) | F◦ N | dν (F)= dσ (θ)dν (F)
ω n,k hk (θ) F n,k
ZGn,k k ZGn,kZSF PF(KN)
1
= dσ (θ)dν (F)
hk (θ) F n,k
ZGn,kZSF KN
k/n
1
6 dσ (θ)dν (F)
hn (θ) F n,k
ZGn,kZSF KN !
1 k/n
= dσ (θ)
(cid:18)ZSn−1 hnKN(θ) n (cid:19)
k/n
K
= | N◦| .
ω
(cid:18) n (cid:19)
Apply now the Blaschke-Santalo´ inequality and the fact that K c Z (µ) (with probability greater
N 7 logN
⊇
than 1 exp( c√N) (notice that logN logN/n for the range of N we use)) to get
− − ≃
k/n k/n k/n
K ω ω
(3.5) | N◦| 6 n 6 n .
ω K c Z (µ)
(cid:18) n (cid:19) (cid:18)| N|(cid:19) (cid:18)| 7 logN |(cid:19)
Since logN is greater than √n we can apply the inequality Z (K)1/n >cL 1 (logN)/n to arrive at
| logN | −µ
p
k
P (K ) c L
(3.6) | F◦ N |dν (F)6 8 µ .
n,k
ω √logN
ZGn,k k (cid:18) (cid:19)
Finally, we apply Markov’sinequality and the reverse Santalo´ inequality of Bourgainand V. Milman [10] to
complete the proof. ✷
10