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On the volume of caps and bounding the
mean-width of an isotropic convex body
Accepted for publication in Math. Proc. Cambridge Phil. Soc.
Peter Pivovarov∗
March 12th, 2010
Abstract
Let K be a convex body which is (i) symmetric with respect to each of
the coordinate hyperplanes and (ii) in isotropic position. We prove that
most linear functionals acting on K exhibit super-Gaussian tail behavior.
Using known facts about the mean-width of such bodies, we then deduce
strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound
its mean-width.
1
Introduction
Let K be a convex body in Rn with volume vol (K) = 1 and suppose
its center of mass is the origin. As is commonly done in the asymptotic
theory of convex bodies, we treat K as a probability space. In particular,
for each unit vector θ, we view the linear functional h·, θi : K → R given
by
hx, θi = x1 θ1 + . . . + xn θn , (x ∈ K),
as a random variable on K. Motivated by Bourgain’s approach to the
Slicing Problem [4], recent research has focused on the distribution of
the functionals h·, θi. In particular, efforts have been made to show that
for any such K, there exists a direction θ which exhibits sub-Gaussian
tail-decay, meaning that,
2
vol {x ∈ K : |hx, θi| > t kh·, θik2 } ≤ e−ct
(1)
2
for
R all t 2≥ 1, where c > 0 is an absolute constant and kh·, θik2 =
|hx, θi| dx. The papers [7] and [15] contain the latest developments, as
K
well as further motivation and history. In this article, we consider bounds
∗ This article is part of the author’s Ph.D. thesis, written under the supervision of Professor
Nicole Tomczak-Jaegermann at the University of Alberta. The author holds an Izaak Walton
Killam Memorial Scholarship.
1
involving the reverse inequality, namely super-Gaussian estimates of the
form
2
vol {x ∈ K : |hx, θi| > t kh·, θik2 } ≥ e−ct ,
(2)
for t > 0 in some suitable range. Such estimates are garnering increased
attention, as in [14] and [21], and are the starting point for this paper.
Our first main result concerns super-Gaussian directions for convex
bodies that are isotropic and 1-unconditional. By isotropic, we mean that
K has volume one, center of mass at the origin and each functional has
the same variance, i.e.,
Z
hx, θi2 dx = L2K for each θ ∈ S n−1 ,
(3)
K
where LK is the isotropic constant of K. Any convex body has an affine
image which is isotropic and LK is an affine-invariant (see, e.g., the survey
[19]). By 1-unconditional, we mean that for each x = (x1 , . . . , xn ) ∈ K,
the parallelepiped [− |x1 | , |x1 |] × . . . × [− |xn | , |xn |] is contained in K. For
such bodies, there are many super-Gaussian directions, which we gauge
in terms of the Haar measure σ on the sphere S n−1 .
Proposition 1.1. There exists an absolute constant C ≥ 1 such that for
any integer n ≥ 1, and any 1-unconditional isotropic convex body K ⊂ Rn ,
the σ-measure of the set of θ ∈ S n−1 such that
vol ({x ∈ K : |hx, θi| ≥ t}) ≥ exp(−Ct2 )
whenever
C≤t≤
√
n
,
C log n
(4)
(5)
is at least 1 − 2−n .
The isotropic constant LK = kh·, θik
√ from the
√ 2 has been omitted
statement since such bodies satisfy 1/ 2πe ≤ LK ≤ 1/ 2 (see, e.g.,
[3]). Proposition 1.1 complements results of Bobkov and Nazarov [2], who
treated the case of sub-Gaussian directions.
As a corollary, we get lower bounds for the volume of caps defined in
terms of the width of the body, measured in terms of the support function
hK (θ) := supx∈K hx, θi.
e = C(β)
e
Corollary 1.2. Let β > 0. Then there is a constant C
such
that for any integer n ≥ 1, and any 1-unconditional isotropic convex body
K ⊂ Rn , the σ-measure of the set of θ ∈ S n−1 satisfying
whenever
e 2 n log n)
vol ({x ∈ K : |hx, θi| ≥ εhK (θ)}) ≥ exp(−C Cε
q
C
e log n
Cn
≤ε≤
C
p
1
e log3/2 n
C
(6)
(7)
e = 3(β + 1) and C is the constant from
is at least 1 − 2−n − 2n−β , where C
Proposition 1.1.
2
For us, the motivation for bounding the volume of the caps in (6) comes
from a paper by Giannopoulos and Milman [10] involving approximation
of a convex body by a random polytope. Corollary 1.2 shows that for
1-unconditional bodies, one has better estimates in most directions. Such
estimates, in turn, are intimately related to mean-width, which leads us
to the second purpose of this paper.
Finding the correct upper bound for the mean-width of an arbitrary
isotropic convex body (not necessarily 1-unconditional) is a problem wellknown to specialists. It has numerous connections and implications (some
of which we review below). We connect the latter problem with volume
estimates for caps, similar to (6), and give a sufficient condition under
which one can bound the mean-width. Our approach may be of independent interest since it involves approximating a convex body by a random
polytope with relatively few vertices.
The paper is organized as follows. The proofs of Proposition 1.1 and
Corollary 1.2 are in §2, the first three subsections of which point out the
key ingredients. The observations about mean-width are contained in §3,
the main result being Proposition 3.4.
Lastly, a few words on notation and viewpoint. Our results are most
meaningful when the dimension n is large. Throughout, c, c1 , C, C ′ , . . .,
etc. denote absolute constants (in particular, independent of n and K).
The symbol |·| will serve the dual role of the standard Euclidean norm
on Rn and also the absolute value of a scalar, the use of P
which will be
clear from the context; for x = (x1 , . . . , xn ) ∈ Rn , kxk1 = n
i=1 |xi | and
kxk∞ = maxi≤n |xi |.
2 Super-Gaussian estimates in 1-unconditional
isotropic convex bodies
We begin by isolating the key ingredients in the proof of Proposition 1.1.
Our proof generalizes an argument due to Schmuckenschläger [27, Proposition 3.4], who showed that the diagonal direction θd = √1n (1, 1, . . . , 1) is
super-Gaussian for the unit ball of ℓn
p for 1 ≤ p ≤ ∞. Our first step is to
pass from θd to a large subset of directions with “well-spread” coordinates.
2.1
Well-spread vectors on the sphere
The next lemma identifies the set of directions for which we will establish
estimate (4) in Proposition 1.1. Similar facts have been used in various
problems (e.g., the use of “incompressible” vectors as in [26]). We include
a proof for completeness.
Throughout, we use the following notation
[n] := {1, . . . , n}.
(8)
Lemma 2.1. There exist absolute constants C1 > 0 and κ > 0 such that
3
for any integer n ≥ 1, the set
Θ := θ ∈ S n−1 |∃I = I(θ) ⊂ [n] with #I ≥ κn :
1
√
C1
C1
≤ |θi | ≤ √ ∀ i ∈ I
n
n
(9)
has σ-measure at least 1 − 2−n .
For the proof, we will use the following standard facts.
Lemma 2.2. There exists an absolute constant c′ > 0 such that for any
integer n ≥ 1, the set
√
√
Θ′ := {θ ∈ S n−1 : c′ n ≤ kθk1 ≤ n}
(10)
has σ-measure at least 1 − 2−n .
Lemma 2.2 follows from, e.g., [20, §2.3 & §5.3] or [25, Theorem 6.1].
The second fact we need is the Paley-Zygmund inequality.
Lemma 2.3. Let Z be a non-negative random variable with finite variance. Then for each t ∈ (0, 1),
P (Z ≥ tEZ) ≥ (1 − t)2
(EZ)2
.
EZ 2
(11)
For a proof, see, e.g., [16, Lemma 0.2.1 ].
Proof of Lemma 2.1. Fix θ ∈ Θ′ (from Lemma 2.2) and write αi :=
√
|θi | n. Without loss of generality, we may assume that the αi are distinct. Let Z be a random variable such that P (Z = αi ) = 1/n. Then
EZ =
n
n
1X
1 X
|θi |
αi = √
n i=1
n i=1
and hence c′ ≤ EZ ≤ 1. By Markov’s inequality, for any λ > 0, we have
P (Z > λ) ≤ P (Z > λEZ) ≤
1
λ
and hence we obtain
#{i ∈ [n] : αi ≤ λ} ≥ (1 − λ−1 )n.
(12)
Next, observe that
EZ 2 =
n
n
1X 2 X 2
αi =
θi = 1.
n i=1
i=1
By the Paley-Zygmund inequality (Lemma 2.3), we have
P Z ≥ c′ /2 ≥ P (Z ≥ (1/2)EZ) ≥ (c′ )2 /4.
and therefore
#{i ∈ [n] : αi ≥ c′ /2} ≥ (c′ )2 n/4.
By (12) and (13), we conclude the result.
4
(13)
2.2
Main probabilistic ingredients
The proof of Proposition 1.1 relies on two theorems about independent
Rademacher random variables ε1 , . . . , εn , i.e.,
P (εi = 1) = P (εi = −1) = 1/2,
i = 1, . . . , n,
(14)
which we state here for the reader’s convenience. The first is the Contraction Principle; see, e.g., [17, Theorem 4.4].
Theorem 2.4. Let ε1 , . . . , εn be independent Rademacher random variables. Let x1 , . . . , xn be elements of a Banach space B and let α1 , . . . , αn
be real numbers such that |αi | ≤ 1 for all i = 1, . . . , n. Then for any t > 0,
n
n
!
!
X
X
P αi εi xi > t ≤ 2P εi xi > t .
(15)
i=1
i=1
The second ingredient is the following theorem about super-Gaussian
estimates for Rademacher sums, which can be found in [17, §4.1].
Theorem 2.5. There is an absolute constant C2 ≥ 1 such that if ε1 , . . . , εn
are independent Rademacher random variables (as in (14)) and if s ∈ R
and ξ ∈ Rn satisfy
|ξ|2
C2 |ξ| ≤ s ≤
,
(16)
C2 kξk∞
then
!
n
X
P
εi ξi ≥ s ≥ exp(−C2 s2 / |ξ|2 ).
(17)
i=1
To show that each θ ∈ Θ (Lemma 2.1) satisfies the super-Gaussian
estimate (4), Theorem 2.4 will be used to pass to subspaces E(I) :=
span{ei : i ∈ I} on which we have control of the coordinates of θ. To use
Theorem 2.5, we will need volume estimates for certain sets involving the
|·| and k·k∞ norms on the orthogonal projection of K onto E(I). This is
done in the next section.
2.3
Projections and retention of volume
Here we prove a lemma which gives a uniform lower bound for the volume
of certain sets that will be used in conjunction with Theorems 2.4 and
2.5. We emphasize that it is a general fact, true for arbitrary isotropic
convex bodies and arbitrary subspaces (not just unconditional bodies and
coordinate subspaces as we need here).
For 1 ≤ ℓ ≤ n, let Gn,ℓ denote the set of all ℓ-dimensional subspaces
of Rn ; for E ∈ Gn,ℓ , let PE be the orthogonal projection onto E.
Lemma 2.6. There exist positive absolute constants C ′ , C ′′ and c such
that for each integer n ≥ 1, for any isotropic convex body K ⊂ Rn , any
ℓ ∈ [n] and any E ∈ Gn,ℓ , the intersection of the sets
√
√
′
KE
:= {x ∈ K : (1/C ′ ) ℓLK ≤ |PE x| ≤ C ′ ℓLK }
(18)
and
say KE :=
′
KE
′′
KE
:= {x ∈ K : kPE xk∞ ≤ C ′′ LK log n},
∩
′′
,
KE
has volume greater than c.
5
(19)
The proof relies on two basic facts. See, for instance, [8, Proposition
2.5.1] and [8, Proposition 2.1.1].
Fact 2.7. There exists an absolute constant C3 such that if n ≥ 1, K ⊂ Rn
is an isotropic convex body and N is a finite subset of the Euclidean ball
B2n , then
Z
max |hx, θi| dx ≤ C3 LK log(#N ).
K θ∈N
Fact 2.8. There exists an absolute constant C4 such that if n ≥ 1, K ⊂ Rn
is a convex body of volume one, and if f : Rn → R is a semi-norm, then
Z
f p (x)dx
K
1/p
≤ C4 p
Z
f (x)dx
K
for all p ≥ 1.
(20)
Proof of Lemma 2.6. Let K ⊂ Rn be an isotropic convex body, ℓ ∈ [n]
and E ∈ Gn,ℓ . Then
Z
K
|PE x| dx ≤
Z
K
|PE x|2 dx
1/2
=
√
ℓLK .
By Markov’s inequality, for any τ > 0, we have
√
1
vol {x ∈ K : |PE x| ≤ τ ℓLK } ≥ 1 − 2 .
τ
(21)
Setting c1 := (2C4 )−1 , where C4 is the constant from Fact 2.8, we have
Z
K
|PE x| dx ≥ c1
Z
K
|PE x|2 dx
1/2
√
= c1 ℓLK .
Applying the Paley-Zygmund inequality (Lemma 2.3), we get
√
vol {x ∈ K : |PE x| ≥ (c1 /2) ℓLK } ≥ c21 /4.
(22)
Taking into account (21) and (22), we determine that there are positive
absolute constants C ′ and c > 0 for which
√
√
′ vol KE
= {x ∈ K : (1/C ′ ) ℓLK ≤ |PE x| ≤ C ′ ℓLK } ≥ 2c.
(23)
To conclude, set C ′′ := C3 /c, where C3 is the constant from Fact 2.7.
Since
kPE xk∞ = max |hPE x, ei i| = max |hx, PE ei i| ,
i≤n
i≤n
we can apply Markov’s inequality and Fact 2.7 to obtain
′′ vol KE
= vol x ∈ K : kPE xk∞ ≤ C ′′ LK log n ≥ 1 − c.
Thus
′
′′ vol (KE ) = vol KE
∩ KE
≥ c,
which concludes the proof.
6
2.4
Proof of Proposition 1.1
Here we combine the results of the previous sections to complete the proof.
Proof of Proposition 1.1. Assume K is a 1-unconditional isotropic convex
body in Rn . Consider C1 , κ and Θ from Lemma 2.1. Set ℓ := ⌊κn⌋, the
largest integer less than κn. Fix θ ∈ Θ so that
C1
1
√ ≤ |θi | ≤ √
C1 n
n
for all i ∈ I, where I = I(θ) ⊂ [n] and |I| = ℓ. Set
E(I) := span{ei : i ∈ I},
n
where the ei ’s are the standard unit vector basis for R
. By Lemma 2.6,
′
′′
the intersection KE(I) ∩ KE(I) has volume vol KE(I) ≥ c.
Let X = (x1 , . . . , xn ) be a random vector distributed uniformly in K.
Let ε1 , . . . , εn be independent Rademacher random variables (cf. (14)).
Then X and (ε1 X1 , . . . , εn Xn ) have the same distribution. Denote the
probability measure corresponding to X, namely vol (·|K ), by PK ; by Pε
the product-measure corresponding to ε = (ε1 , . . . , εn ). Then
n
!
X
vol ({x ∈ K : |hx, θi| > t}) = PK θi xi > t
i=1
n
!
X
= PK ⊗ Pε εi θi xi > t
i=1
n
!
Z
X
=
Pε εi θi xi > t dx
K
i=1
!
Z
X
≥ (1/2)
Pε εi θi xi > t dx (by Thm. 2.4)
K
i∈I
!
Z
X
≥ (1/2)
Pε εi θi xi > t dx.
(24)
KE(I)
i∈I
Fix x ∈ KE(I) , and set y = (θi xi )i∈I . Then, by definition of KE(I) and
Θ,
C1 C ′′ LK log n
C1 √
kyk∞ ≤ √ PE(I) x∞ ≤
n
n
and
√
√
1 C1 C1 C ′ ℓLK
ℓLK
√
√
√
√
≤
.
P
x
≤
|y|
≤
P
x
≤
E(I)
E(I)
C1 C ′ n
C1 n
n
n
√
Since K is 1-unconditional, LK ≤
√ 1/ 2, (e.g., [3]). Moreover, for any
convex body K, LK ≥ LB2n ≥ 1/ 2πe. Recalling that ℓ = ⌊κn⌋, we
conclude that there exist absolute constants A1 > 1 and A2 > 1 such that
kyk∞ ≤
A1 log n
√
n
7
and
1
≤ |y| ≤ A2 .
A2
Let C2 be the constant from Theorem 2.5. At this point we can determine the constant C asserted in Proposition 1.1: take C := A1 A22 C2 .
Then our assumption (5) implies
C2 |y| ≤ t ≤
|y|2
,
C2 kyk∞
making (24) ripe for an application of Theorem 2.5:
!
Z
Z
X
(1/2)
Pε εi θi xi > t dx ≥ (1/2)
exp(−C2 t2 / |y|2 )dx
KE(I)
KE(I)
i∈I
≥ (c/2) exp(−C2 A22 t2 )
≥ (c/2) exp(−Ct2 ),
2
where we have used the notation y = (θi xi )i∈I as above. Since c/2 ≥ e−t
for t large enough, we can recover the proposition as stated simply by
adjusting the constants.
Remark 2.9. The idea in the proof of Proposition 1.1 was recently adapted
and used in [5, Lemma 2.7].
2.5
Proof of Corollary 1.2
To prove Corollary 1.2, we will need two additional results.
Lemma 2.10. For any M ∈ (0, 1), the set
Θ1 := {θ ∈ S n−1 : kθk∞ ≤ M }
has σ-measure at least 1 − 2ne−nM
2
/2
.
Proof. Using the well-known estimate
σ(θ ∈ S n−1 : |he1 , θi| > M ) ≤ 2e−nM
2
/2
,
(25)
(see, e.g., [1, Lemma 2.2]), we have
σ θ ∈ S n−1 : ∃i ≤ n : |hei , θi| > M ≤ nσ θ ∈ S n−1 : |he1 , θi| > M
≤ 2ne−nM
2
/2
.
Another result, due to Bobkov and Nazarov [3, Propositions 2.4 &
2.5], will also be of use.
Proposition 2.11. Let K be a 1-unconditional isotropic convex body in
Rn . Then
p
√
√
[−LK / 2, LK / 2]n ⊂ K ⊂ 3/2nB1n .
In fact, we will use only the right-most inclusion.
8
Proof of Corollary 1.2. Let β > 0. Apply Lemma 2.10 with Mn :=
q
2(β+1) log n
so that σ(Θ1 ) ≥ 1 − 2n−β . By Proposition 2.11,
n
hK (θ) ≤
n−1
p
3/2n kθk∞
for each θ ∈ S
. Thus
p
σ θ ∈ S n−1 : hK (θ) ≤ 3(β + 1)n log n ≥ σ(Θ1 )
≥ 1 − 2n−β .
Let Θ be the set from Lemma 2.1. As the proof of Proposition 1.1 shows,
any element of Θ satisfies the super-Gaussian estimate (4). Thus if θ ∈
Θ ∩ Θ1 , we have
p
vol ({x ∈ K : |hx, θi| ≥ εhK (θ)}) ≥ vol {x ∈ K : |hx, θi| ≥ ε 3(β + 1)n log n}
e 2 n log n),
≥ exp(−C Cε
e = 3(β + 1)) provided that
(where C
p
C ≤ ε 3(β + 1)n log n ≤
√
n
,
C log n
where C is the constant from Proposition 1.1.
3 On the mean-width of an isotropic convex body
For a convex body K ⊂ Rn , denote its support function by
hK (θ) := sup hx, θi ,
x∈K
(θ ∈ S n−1 ).
The width of K in the direction of θ is the quantity w(K, θ) = hK (θ) +
hK (−θ) and the mean-width of K is
Z
Z
w(K) =
w(K, θ)dσ(θ) = 2
hK (θ)dσ(θ).
S n−1
S n−1
Suppose now that vol (K) = 1. Urysohn’s inequality (see, e.g., [25, Corol√
lary 1.4]) implies that w(K) ≥ c n, with c > 0 an absolute constant.
On the other hand, a combination of results due to Figiel and TomczakJaegermann [6] and Pisier [24], implies that there exists an affine image
T K of K of volume one such that
√
w(T K) ≤ C n log n,
where C is an absolute constant. There is an important position associated
to the latter fact, namely ℓ-position, see, e.g., the survey [11, §2.3], for a
discussion of the corresponding circle of ideas.
9
In this section, we discuss upper bounds for the mean-width of a convex
body in isotropic position (as defined in (3)). A result known to specialists
is that for any isotropic convex body K ⊂ Rn , one has
w(K) ≤ Cn3/4 LK ,
(26)
where C is an absolute constant. The latter estimate follows from Dudley’s
entropy estimate as in [9, Theorem 5.6] and the covering number bound
from [18, Lemma 4]; a proof is in [12]. The bound (26) can also be derived
easily using more recent tools, namely results of Paouris on Lq -centroid
bodies in [23] (see also [7, §2 (in particular, (2.2) and Lemma 2.5)].
For the benefit of non-specialists, we mention also that sub-Gaussian
estimates such as (1) in our introduction have implications for the width
of K. In particular, for θ ∈ S n−1 , define
Z
kh·, θikψ2 := inf s > 0 :
exp |hx, θi|2 /s2 dx ≤ 2 .
K
One can check that h·, θi satisfies (1) if and only if kh·, θikψ2 ≤ C1 kh·, θik2 ,
where C1 is an absolute constant. From [22, Lemma 4.2], we have
√
max{hK (θ), hK (−θ)} ≤ C n kh·, θikψ2 .
Thus if one could show that “most” directions are sub-Gaussian (or nearly
sub-Gaussian), then one would obtain a better bound on the mean-width.
As [7] and [15] show, however, it is non-trivial to establish even the existence of one θ which exhibits sub-Gaussian tail-decay.
In this section, we offer another condition, related to lower bounds for
caps similar to (6), under which one can bound the mean-width.
3.1 Bounding the mean-width via random polytopes
Throughout this section, we assume that K is an isotropic convex body
in Rn (as in (3)), X1 , . . . , XN are independent random vectors distributed
uniformly in K; KN their convex hull:
KN := conv {X1 , . . . , XN } ;
P the associated product measure on ⊗N
i=1 K.
(27)
√
Lemma 3.1. Let t ≥ 1 and suppose that n < N ≤ e nt/2 . Then
√
p
P w(KN ) ≤ C log N LK t ≥ 1 − e− nt/2 ,
(28)
where C > 0 is an absolute constant.
Proof. Let u1 , . . . , uN be points on the sphere S n−1 . Then, using (25) in
a standard way, we have
√
Z
C1′ log N
√
max |hui , θi| dσ(θ) ≤
,
(29)
n
S n−1 i≤N
where C1′ is an absolute constant.
10
v(θ, ε)
hK (θ)
εhK (θ)
θ
0
K
Figure 1: The volume v(θ, ε) of a cap.
By [23, Theorem 1.1], we have
√
√
P |Xi | ≤ C2′ nLK t for each i = 1, . . . , N ≥ 1 − e− nt/2 ,
√
where C2′ is an absolute constant. Assume now that 0 < |Xi | ≤ C2′ nLK t
′
and write Xi = Xi / |Xi |. Then
Z
w(KN ) ≤ 2
max |hXi , θi| dσ(θ)
i≤N
S n−1
√
≤ 2C2′ nLK t
≤C
p
Z
S n−1
log N LK t,
max Xi′ , θ dσ(θ)
i≤N
where we used (29) for the last inequality and C = 2C1′ C2′ .
Remark 3.2. See [5, Proposition 3.3] for further observations about the
mean-width of the random polytope KN ; in particular, the relation to the
width of Lq -centroid bodies.
Next, we use an idea of Giannopoulos and Milman from [10, Lemma
5.1]. For each ε ∈ (0, 1) and θ ∈ S n−1 , let
v(θ, ε) := vol ({x ∈ K : hx, θi ≥ εhK (θ)}) ,
as in Figure 3.1.
Lemma 3.3. Let ε > 0. Then
P (hKN (θ) < εhK (θ)) ≤ exp (−N v(θ, ε)) .
Proof. By definition,
vol ({x ∈ K : hx, θi < εhK (θ)}) = 1 − v(θ, ε),
hence
P max hXj , θi < εhK (θ) = (1 − v(θ, ε))N ≤ exp(−N v(θ, ε)).
j≤N
11
(30)
3.2 Sufficient conditions for bounding the meanwidth
In this section, we prove that one can bound the mean-width of an
isotropic convex body under a certain hypothesis; namely, that in “most”
directions θ, the volume of the caps v(θ, ε) (cf. (30)) is suitably large.
“Most” in this case is meant with respect to the Haar measure σ on the
sphere S n−1 , and is quantified by a certain constant; expressly, let C0 be
the smallest constant such that for any positive integer n and any isotropic
convex body K ⊂ Rn ,
Z
S n−1
max |hx, θi|
x∈K
2
dσ(θ)
!1/2
≤ C0
Z
max |hx, θi| dσ(θ),
S n−1 x∈K
(31)
By Fact 2.8, C0 is an absolute constant. It will play a role in the formulation of the proposition.
Proposition 3.4. Let n be a positive integer and K an isotropic convex
body in Rn . Let α ≥ 1, ε ∈ (0, 1) and p ∈ [1, 2]. Let v(θ, ε) be the volume
√
of the cap defined in (30) and C0 as in (31). If 4αεp n ≥ 1 and
p
1
,
(32)
σ {θ ∈ S n−1 : v(θ, ε) ≥ e−αε n } ≥ 1 −
16C02
then
b 3/2 ε3p/2−1 nLK ,
w(K) ≤ Cα
(33)
b is an absolute constant.
where C
Before proving the proposition, we give several remarks to illustrate
its potential utility and emphasize the important ranges for α, ε and p.
Remark 3.5. The argument from [10, Lemma 5.1] shows that for every
θ ∈ S n−1 and every ε ∈ (0, 1), one has
v(θ, ε) ≥
c
(1 − ε)n ,
n2
where c > 0 is an absolute constant. But (c/n2 )(1 − ε)n ≥ e−3εn provided
that log(n2 /c)/n ≤ ε ≤ 1/2. Hence (32) holds with α = 3, ε = n−1/2 ,
and p = 1, in which case the proposition recovers the known estimate:
w(K) ≤ Cn3/4 LK ,
with C an absolute constant.
Remark 3.6. If (32) holds with
α = C4′ log n,
ε=
1
,
n1/4 log1/2 n
one would obtain the optimal bound
p
w(K) ≤ C n log nLK ,
where C is an absolute constant.
12
p = 2,
(34)
Remark 3.7. Corollary 1.2 shows that (32) holds with α, ε and p as in
the previous remark (34) for all 1-unconditional isotropic convex bodies
(with a stronger measure estimate). Note, however, that we have used
Proposition 2.11 (in particular, the upper-bound on the width) to prove
Corollary 1.2. Nevertheless, this shows that (32) holds with the values in
(34) for a large class of convex bodies.
√
Proof of Proposition
3.4. Let t = 4αεp n so that (by assumption) t ≥ 1.
√
Set N = e nt/2 and suppose that X1 , . . . , XN are independent random
vectors distributed uniformly in K and, as in (27), KN is their convex
hull. By Lemma 3.1, we have
p
w(KN ) ≤ C log N LK t
(35)
√
with probability at least 1 − e− nt/2 .
On the other hand, we can use Lemma 3.3 and an approximation
argument, as in [10, Theorem 5.2], to bound the width of K by that of
KN . For convenience, denote the set appearing in (32) by A(α, ε, p). A
standard volume argument shows that for any η ∈ (0, 1), there exists an
η-net N ⊂ A(α, ε, p), i.e., a finite set satisfying the condition
∀θ ∈ A(α, ε, p), ∃θ0 ∈ N such that |θ − θ0 | < η,
with cardinality #N ≤ (3/η)n . In particular, for η = ε/4(n + 1), let us
fix one such η-net N ⊂ A(α, ε, p) with cardinality
#N ≤ (12(n + 1)/ε)n .
(36)
Claim 3.8.
ε
P ∃θ ∈ A(α, ε, p) : hKN (θ) < hK (θ) ≤ P (∃θ0 ∈ N : hKN (θ0 ) ≤ εhK (θ0 )) .
2
(37)
Proof of Claim 3.8. Suppose that there exists θ ∈ A(α, ε, p) such that
hKN (θ) < (ε/2)hK (θ).
Choose θ0 ∈ N such that |θ − θ0 | < η. Note that
LK B2n ⊂ K ⊂ (n + 1)LK B2n ;
(38)
see [13, Theorem 4.1]. The claim then follows from
hKN (θ0 ) ≤ hKN (θ) + hKN (θ0 − θ)
≤ (ε/2)hK (θ) + hK (θ0 − θ)
≤ (ε/2)hK (θ0 ) + (ε/2)hK (θ − θ0 ) + hK (θ0 − θ)
≤ (ε/2)hK (θ0 ) + 2(n + 1)LK η
≤ (ε/2)hK (θ0 ) + 2(n + 1)ηhK (θ0 )
≤ εhK (θ0 )
13
(by (38))
(by (38))
(η = ε/(4(n + 1))).
Claim 3.8 and Lemma 3.3 yield
P ∃ θ ∈ A(α, ε, p) : hKN (θ) <
ε
hK (θ) ≤ #N max exp(−N v(θ0 , ε))
θ0 ∈N
2
n
p
12(n + 1)
≤
exp −eαε n/2 .
ε
At this point a remark on the possible range of ε is in order. Our desired
conclusion (33) is a triviality if α3/2 ε3p/2−1 > 1 (by the diameter bound
(38)); hence we may assume α3/2 ε3p/2−1 ≤ 1, in which case our assump√
tion 4αεp n ≥ 1 yields the restriction ε ≥ 1/(8n3/4 ). Thus the latter
probability is at most
√
p
(96(n + 1))2n exp −eαε n/2 ≤ exp 2n log(96(n + 1)) − e n/8
√
≤ exp −(1/2)e n/8 ,
provided that n satisfies 2n log(96(n + 1)) ≤ (1/2)e
√
n/8
. Therefore
hK (θ) ≤ 2ε−1 hKN (θ) for each θ ∈ A(α, ε, p)
√
with probability at least 1 − exp −e n/8 /2 .
(39)
Thus if KN satisfies both (35) and (39), we have
Z
Z
hK (θ)dσ(θ) ≤ 2ε−1
hKN (θ)dσ(θ)
A(α,ε,p)
A(α,ε,p)
≤ 2ε−1 w(KN )
p
≤ 2ε−1 C log N LK t.
While on the compliment A(α, ε, p)c = S n−1 \A(α, ε, p),
Z
A(α,ε,p)c

hK (θ)dσ(θ) ≤ 
Z
S n−1
≤ C0 w(K)
max |hx, θi|
x∈K
p
2
1/2
dσ(θ)
p
σ(A(α, ε, p)c )
σ(A(α, ε, p)c )
≤ w(K)/4.
Combining the latter estimates,
Z
Z
w(K) = 2
hK (θ)dσ(θ) + 2
A(α,ε,p)
≤ 4ε−1 C
hence
p
hK (θ)dσ(θ)
A(α,ε,p)c
log N LK t + w(K)/2,
b 3/2 ε3p/2−1 nLK ,
w(K) ≤ Cα
b an absolute constant.
with C
14
Acknowledgements
I thank my supervisor, Professor Nicole Tomczak-Jaegermann, for valuable feedback and continued guidance throughout my Ph.D. program.
Most of the research for this article took place while I was visiting the
University of Athens, from February to May 2008, as an Early-Stage
Researcher with the Phenomena in High Dimensions (PHD) European
network (MRTN-CT-2004-511953). I am grateful to the institution, the
PHD network, and, especially, my host Professor Apostolos Giannopoulos for his gracious hospitality and help with all things mathematical and
otherwise. I thank Radek Adamczak for help with some of the technical details in the proof of Proposition 1.1; Grigoris Paouris for helpful
comments. Lastly, I gratefully acknowledge the financial support of the
Killam Trusts.
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Peter Pivovarov
Department of Mathematical and Statistical Sciences,
632 CAB, University of Alberta
Edmonton, AB, Canada
T6G-2G1
[email protected]
17