Estimates for the affine and dual affine quermassintegrals of convex bodies

18 Pages·2011·0.23 MB·English

Estimates for the affine and dual affine
quermassintegrals of convex bodies
Nikos Dafnis and Grigoris Paouris
Abstract
We provide estimates for suitable normalizations of the affine and dual
affine quermassintegrals of a convex body K inRn. These follow by a more
general study of normalized p-means of projection and section functions of
K.
1 Introduction
The starting point of this paper is an integral formula of Furstenberg and Tzkoni
[5] about the volume of k-dimensional sections of ellipsoids: for every ellipsoid E in
Rn and every 16k 6n one has
Z
(1.1) |E ∩F|ndν (F)=c |E|k,
n,k n,k
Gn,k
where ν is the Haar measure on the Grassmannian G and c is a constant
n,k n,k n,k
depending only on n and k; more precisely, c = Γ(cid:0)n +1(cid:1)k(cid:14)Γ(cid:0)k +1(cid:1)n. It
n,k 2 2
was proved by Miles [16] that this formula can be obtained in a simpler way as a
consequence of classical formulas of Blaschke and Petkantschin.
Later, analogous quantities were considered by Lutwak and Grinberg in the
setting of convex bodies. Lutwak introduced in [11] – for every convex body K in
Rn and every 16k 6n−1 – the quantities
!−1/n
ω Z
(1.2) Φ (K)= n |P (K)|−ndν (F) ,
n−k ω F n,k
k Gn,k
where P (K) is the orthogonal projection onto F and ω is the volume of the
F k
Euclidean unit ball in Rk. For k = 0 and k = n one sets Φ (K) = |K| and
0
Φ (K) = ω respectively. Grinberg [8] proved that these quantities are invariant
n n
undervolumepreservingaffinetransformations;thisjustifiestheterminology“affine
quermassintegrals” for Φ (K). From the definition of Φ (K) it is clear that
n−k n−k
ω Z
(1.3) Φ (K)6 n |P (K)|dν (F)=W (K),
n−k ω F n,k n−k
k Gn,k
1
where W (K) = V(K,[k]Bn,[n−k]) are the Quermassintegrals of K. Lutwak
n−k 2
conjectured in [12] that the affine quermassintegrals satisfy the inequalities
(1.4) ωjΦn−j 6ωiΦ (K)n−i
n i n j
for all 06i<j <n. For example, Lutwak asks if
(1.5) Φ (K)>ω(n−k)/n|K|k/n
n−k n
with equality if and only if K is an ellipsoid; note that the weaker inequality
W (K) > ω(n−k)/n|K|k/n holds true by the isoperimetric inequality. Most of
n−k n
these questions remain open (see [6, Chapter 9]); two cases of (1.5) follow from
classical results: when k = n−1 this inequality is the Petty projection inequality
and when k =1 and K is symmetric then (1.5) is the Blaschke-Santal´o inequality.
Lutwak proposed in [13] to study the dual affine quermassintegrals Φ˜ (K).
n−k
For every convex body K in Rn and every 16k 6n−1 one defines
!1/n
ω Z
(1.6) Φ˜ (K)= n |K∩F|ndν (F) .
n−k ω n,k
k Gn,k
Fork =0andk =nonesetsΦ˜ (K)=|K|andΦ˜ (K)=ω respectively. Grinberg
0 n n
provedin[8]thatthesequantitiesarealsoinvariantundervolumepreservinglinear
transformations, and he established the inequality
(1.7) Φ˜ (K)6ω(n−k)/n|K|k/n
n−k n
for all 1 6 k 6 n−1, with equality if and only if K is a centered ellipsoid. The
case k =n−1 of this inequality is the Busemann intersection inequality (while the
case k =1 becomes an identity for symmetric convex bodies).
Being affinely invariant, affine and dual affine quermassintegrals appear to be
useful in asymptotic convex geometry. So, one of the purposes of this work is to
give upper and lower bounds for Φ (K) and Φ˜ (K) in the remaining cases.
n−k n−k
Weintroduceadifferentnotationandnormalizationwhichisbetteradaptedtoour
needs. Nevertheless, the question we study is equivalent to e.g. [6, Problem 9.7].
Definition1.1(normalizedaffinequermassintegrals). Foreveryconvexbody
K in Rn and every 16k 6n−1 we define
!− 1
Z kn
(1.8) Φ (K)= |P (K)|−ndν (F) .
[k] F n,k
Gn,k
WealsosetΦ (K)=|K|1/n. Lutwak’sconjecturesaboutaffinequermassintegrals
[n]
can now be restated as follows:
(i)Forevery(symmetric)convexbodyK ofvolume1inRnandevery16k 6n−1,
(1.9) Φ (K)>Φ (D ),
[k] [k] n
2
where D is the Euclidean ball of volume 1.
n
(ii) For every convex body K of volume 1 in Rn and every 16k 6n−1,
(1.10) Φ (K)6Φ (S ),
[k] [k] n
where S is the regular Simplex of volume 1.
n
In view of these conjectures, in the asymptotic setting it is reasonable to ask if the
following holds true: There exist absolute constants c ,c > 0 such that for every
1 2
convex body K of volume 1 in Rn and every 16k 6n−1,
p p
(1.11) c n/k 6Φ (K)6c n/k.
1 [k] 2
For k = 1 the Blaschke-Santal´o inequality shows that (1.9) holds true. Proving
(1.10) for k = 1 corresponds to Malher’s conjecture. Clearly, (1.11) for k = 1
follows from the Blaschke-Santal´o and the reverse Santal´o inequality of Bourgain-
Milman [3].
Note that for k =n−1 we have
(cid:18) |Bn| (cid:19)n(n1−1)
(1.12) Φ (K)= 2 ,
[n−1] |Π∗(K)|
where Π∗(K) is the polar projection body of K. Then, H¨older’s inequality and the
isoperimetric inequality show that (1.9) holds true. The same is true for (1.10):
this follows from Zhang’s inequality; see [30].
Definition1.2(normalizeddualaffinequermassintegrals). Foreveryconvex
body K in Rn and every 16k 6n−1 we define
! 1
Z kn
(1.13) Φ˜ (K)= |K∩F⊥|ndν (F) .
[k] n,k
Gn,k
Grinberg’stheoremaboutdualaffinequermassintegralsstatesthatifK hasvolume
1 then
(1.14) Φ˜ (K)6Φ˜ (D )6c ,
[k] [k] n 2
where c > 0 is an absolute constant. As we will see, if the hyperplane conjecture
2
has an affirmative answer then
(1.15) Φ˜ (K)>c
[k] 1
for every centered convex body of volume 1, where c >0 is an absolute constant.
1
Inviewoftheabove, hereoneasksifthefollowingholdstrue: Thereexistabsolute
constants c ,c >0 such that for every centered convex body K of volume 1 in Rn
1 2
and every 16k 6n−1,
(1.16) c 6Φ˜ (K)6c .
1 [k] 2
3
Our estimates on the normalized affine and dual affine quermassintegrals are
summarized in the following:
Theorem 1.3. Let K be a convex body of volume 1 in Rn. Then, for every 1 6
k 6n−1,
p
(1.17) Φ (K)6c n/k logn
[k] 1
and, if K is also centered,
c
(1.18) Φ˜ (K)> 2 ,
[k] L
K
where L is the isotropic constant of K. In particular, assuming the hyperplane
K
conjecture we have that Φ˜ (K)’1 for all 16k 6n−1. We also have the bounds
[k]
p
(1.19) Φ (K)6c (n/k)3/2 log(en/k)
[k] 3
and
c
(1.20) Φ˜ (K)> 4 .
[k] p p
n/k log(en/k)
which are sharp when k is proportional to n.
Fortheproofsoftheseestimates,weattemptamoregeneralstudyofnormalized
p-means of projection and section functions of K, which we introduce for every
16k 6n−1 and every p6=0 by setting
! 1
Z kp
(1.21) W (K):= |P (K)|pdν (F) .
[k,p] F n,k
Gn,k
and
! 1
Z kp
(1.22) W˜ (K)= |K∩F⊥|pdν (F) .
[k,p] n,k
Gn,k
respectively. The k-th normalized affine and dual affine quermassintegrals of K
correspond to the cases p=−n and p=n respectively:
(1.23) Φ (K)=W (K)andΦ˜ (K)=W˜ (K).
[k] [k,−n] [k] [k,n]
We list several properties of the p-means and prove some related inequalities.
Acknowledgment. We would like to thank Apostolos Giannopoulos for many
interestingdiscussions. ThesecondnamedauthorwishestothanktheUSNational
Science Foundation for support through the grant DMS-0906150.
4
2 Notation and Preliminaries
We work in Rn, which is equipped with a Euclidean structure h·,·i. We denote
by k·k the corresponding Euclidean norm, and write Bn for the Euclidean unit
2 2
ball, and Sn−1 for the unit sphere. Volume is denoted by |·|. We write ω for the
n
volume of Bn and σ for the rotationally invariant probability measure on Sn−1.
2
The Grassmann manifold G of k-dimensional subspaces of Rn is equipped with
n,k
the Haar probability measure ν . We also write A for the homothetic image of
n,k
volume 1 of a compact set A ⊆ Rn of positive volume, i.e. A := |A|−n1A. If A
and B are compact sets in Rn, then the covering number N(A,B) of A by B is the
smallest number of translates of B whose union covers A.
Thelettersc,c0,c ,c etc. denoteabsolutepositiveconstantswhichmaychange
1 2
from line to line. Whenever we write a ’ b, we mean that there exist absolute
constants c ,c >0 such that c a6b6c a.
1 2 1 2
A star-shaped body C with respect to the origin is a compact set that satisfies
tC ⊆C for all t∈[0,1]. We denote by k·k the gauge function of C:
C
(2.1) kxk =inf{λ>0:x∈λC}.
C
A convex body in Rn is a compact convex subset C of Rn with non-empty interior.
We say that C is symmetric if x ∈ C implies that −x ∈ C. We say that C is
centered if it has centre of mass at the origin: R hx,θidx = 0 for every θ ∈ Sn−1.
C
Thesupportfunctionh :Rn →RofC isdefinedbyh (x)=max{hx,yi:y ∈C}.
C C
The radius of C is the quantity R(C)=max{kxk :x∈C} and, if the origin is an
2
interior point of C, the polar body C◦ of C is
(2.2) C◦ :={y ∈Rn :hx,yi61for allx∈C}.
Let K be a centered convex body of volume 1 in Rn. Then, the Blaschke–Santal´o
inequality and the Bourgain–Milman inequality imply that
1
(2.3) |K◦|n1 ’ .
n
Let K be a centered convex body in Rn. For every F ∈ G , 1 6 k 6 n−1, we
n,k
have that P (K◦)=(K∩F)◦, and hence,
F
1
(2.4) |K∩F|1/k|P K◦|1/k ’ .
F k
The Rogers-Shephard inequality [26] states that
(cid:18)n(cid:19)1/k en
(2.5) 16|P K|1/k|K∩F⊥|1/k 6 6 .
F k k
We refer to the books [28], [21] and [25] for basic facts from the Brunn-Minkowski
theory and the asymptotic theory of finite dimensional normed spaces.
5
Let K be a centered convex body of volume 1 in Rn. For every q > 1 and
θ ∈Sn−1 we define
(cid:18)Z (cid:19)1/q
(2.6) h (θ):= |hx,θi|qdx .
Zq(K)
K
We define the L -centroid body Z (K) of K to be the centrally symmetric convex
q q
setwithsupportfunctionh . L –centroidbodieswereintroducedin[14]. Here
Zq(K) q
we follow the normalization (and notation) that appeared in [23].
It is easy to check that Z (K)⊆Z (K)⊆Z (K)⊆Z (K) for every 16p6
1 p q ∞
q 6∞, where Z (K)=conv{K,−K}. Note that if T ∈SL(n) then Z (T(K))=
∞ p
T(Z (K)). Moreover, as a consequence of the Brunn–Minkowski inequality (see,
p
for example, [23]), one can check that
q
(2.7) Z (K)⊆c Z (K)
q p p
for all 16p<q, where c>1 is an absolute constant, and
(2.8) Z (K)⊇cK
q
for all q >n, where c>0 is an absolute constant.
A centered convex body K of volume 1 in Rn is called isotropic if Z (K) is a
2
multiple of Bn. Then, we define the isotropic constant of K by
2
(cid:18)|Z (K)|(cid:19)1/n
(2.9) L := 2 .
K |Bn|
2
It is known that L > L > c > 0 for every convex body K in Rn. Bourgain
proved in [2] that LK 6 cB√42nnlogn and, a few years ago, Klartag [9] obtained the
√ K
estimate L 6 c4n (see also [10]). The hyperplane conjecture asks if L 6 C,
K K
where C > 0 is an absolute constant. We refer to [19], [7] and [23] for additional
information on isotropic convex bodies.
Let K be a centered convex body of volume 1 in Rn. For every star shaped
body C in Rn and any −n<p6∞, p6=0, we set
(cid:18)Z (cid:19)1/p
(2.10) I (K,C):= kxkpdx .
p C
K
If C =Bn we simply write I (K) instead of I (K,Bn).
2 p p 2
3 p-mean projection functions and estimates for
Φ (K)
[k]
We first consider the question whether there exist absolute constants c ,c > 0
1 2
such that for every convex body K of volume 1 in Rn and every 16k 6n−1,
p p
(3.1) c n/k 6Φ (K)6c n/k.
1 [k] 2
We can prove that the right-hand side inequality holds true up to a logn term.
6
Theorem 3.1. Let K be a centered convex body of volume 1 in Rn. Then, for
every 16k 6n−1,
p
(3.2) Φ (K)6c n/k logn.
[k]
For the proof of Theorem 3.1 we introduce a normalized version of the quermass-
integrals of a convex body.
§3.1 Normalized quermassintegrals. Let K be a convex body in Rn. For every
16k 6n−1 we define the normalized k-quermassintegral of K by
!1/k
Z
(3.3) W (K):= |P (K)|dν (F) .
[k] F n,k
Gn,k
We also set W (K)=|K|1/n and W (K)=1. Note that
[n] [0]
Z
(3.4) W (K)= [h (θ)+h (−θ)]dσ(θ)=2w(K).
[1] K K
Sn−1
From the definition and Kubota’s formula (see [28]) it is clear that, for every 1 6
k 6n−1 one has
(cid:18)ω (cid:19)1/k
(3.5) W (K)= kV(K,[k];Bn,[n−k]) .
[k] ω 2
n
Applying the Aleksandrov-Fenchel inequality (see [28, Chapter 6]) one can check
the following:
(i) If K and L are convex bodies in Rn, then, for all 16k 6n,
(3.6) W (K+L)>W (K)+W (L).
[k] [k] [k]
(ii) For all 06k <k <k 6n,
1 2 3
(3.7) W[k2](K)W[k1](B2n) >(cid:18)W[k3](K)W[k1](B2n)(cid:19)(kk22(−k3k−1k)k13) .
W (K)W (Bn) W (K)W (Bn)
[k1] [k2] 2 [k1] [k3] 2
(iii) For all 16k 6k 6n,
1 2
W (K) W (K)
(3.8) [k2] 6 [k1] .
W (Bn) W (Bn)
[k2] 2 [k1] 2
Proof of Theorem 3.1. Since Φ (K) is affine invariant we may assume that K
[k]
is centered. It is well-known that Pisier’s inequality (see [25, Chapter 2]) on the
norm of the Rademacher projection implies that there exists T ∈SL(n) such that
√
(3.9) W (T(K))=2w(T(K))6c nlogn.
[1]
7
More precisely, (3.9) follows from Pisier’s inequality in the case where K is sym-
metric. However, it is not difficult to extend the inequality to the non necessarily
symmetric case (see e.g. [22, Lemma3]). Then, using the affine invariance of Φ
[k]
and the fact that Φ (K)6W (K), we write
[k] [k]
(3.10) Φ (K)=Φ (T(K))6W (T(K)).
[k] [k] [k]
Since W (Bn)=ω1/k ’ √1 , it follows from (3.8) that
[k] 2 k k
(3.11) W (T(K))6 W[k](B2n)W (T(K))6cpn/k logn.
[k] W (Bn) [1]
[1] 2
This completes the proof. 2
Next, we introduce the p-mean projection function W (K) and the p-mean
[k,p]
width w (K) of a convex body K and prove a weak lower bound in the direction
p
of the left hand side inequality of (3.1).
§3.2. p-mean projection function. Let K be a convex body in Rn. For every
1 6 k 6 n − 1 and for every p 6= 0 we define the p-mean projection function
W (K) by
[k,p]
! 1
Z kp
(3.12) W (K):= |P (K)|pdν (F) .
[k,p] F n,k
Gn,k
We also set W (K):=|K|1/n. Observe that the k-th normalized affine quermass-
[n]
integral of K corresponds to the case p=−n:
(3.13) Φ (K):=W (K).
[k] [k,−n]
It is clear that W (K) is an increasing function of p, W (λK) = λW (K)
[k,p] [s,p] [s,p]
for every λ > 0 and W (K) 6 W (L) whenever K ⊆ L. Moreover, for every
[s,p] [s,p]
16k <m6n−1 and every p6=0, one has
! 1
Z kp
(3.14) W (K)= Wkp (P (K))dν (E) .
[k,p] [k,p] E n,m
Gn,m
In particular,
!− 1
Z km
(3.15) W (K)= Φ−km(P (K))dν (E) .
[k,−m] [k] E n,m
Gn,m
§3.3. p-mean width. The p-mean width of K is defined for every p6=0 by
(cid:18)Z (cid:19)1/p
(3.16) w (K)= hp (θ)dσ(θ) .
p K
Sn−1
8
It is clear that w (K) is an increasing function of p, w (λK) = λw (K) for every
p p p
λ>0 and w (K)6w (L) whenever K ⊆L. Note that, if K◦ is the polar body of
p p
K, then
(cid:18)|Bn|(cid:19)n1
(3.17) w (K)= 2 .
−n |K◦|
Also, for every 16k 6n−1,
!1/p
Z
(3.18) w (K)= wp(P (K))dν (E)
p p E n,k
Gn,k
and, in particular,
!−1/k
Z
(3.19) w (K)=ω1/k |(P (K))◦|dν (E) .
−k k E n,k
Gn,k
Usingtheaboveweareabletoprovethat,inthesymmetriccase,W (K)>
[k,−q]
cpn/k as far as q 6n/k; recall that Φ (K)=W (K).
[k] [k,−n]
Theorem 3.2. Let K be a symmetric convex body of volume 1 in Rn. Then, for
every 16k 6n−1,
p
(3.20) W (K)>c n/k.
[k,−n/k]
Proof. Using H¨older’s inequality, the Blaschke-Santal´o and the reverse Santal´o
inequality, for every p>1 we can write
! 1 ! 1
Z kp Z |(P (K))◦|p kp
|P (K)|−pdν (F) ’ F dν (F)
F n,k ω2p n,k
Gn,k Gn,k k
√ Z (cid:18)Z 1 (cid:19)p !k1p
’ k dσ (θ) dν (F)
hk (θ) F n,k
Gn,k SF K
! 1
√ Z Z 1 kp
6 c k dσ (θ)dν (F)
Gn,k SF hkKp(θ) F n,k
! 1
√ Z 1 kp
= c k dσ(θ)
Sn−1 hkKp(θ)
√
= c kw−1 (K).
−kp
We set p:=n/k >1. Then, from (3.17) we get
(3.21) W (K)> w−√n(K) ’ √1 ωn1/n ’pn/k.
[k,−n/k] c k c k|K◦|1/n
9
This completes the proof. 2
Note. What we have actually shown in the proof of Theorem 3.2 is that
(3.22)
√ Z (cid:18)Z 1 (cid:19)p !−k1p w (K)
W (K)’ k dσ (θ) dν (F) >c −√kp
[k,−p] hk (θ) F n,k k
Gn,k SF K
for all 16k 6n−1 and p>1.
˜
4 p-mean section functions and estimates for Φ (K)
[k]
Next,weconsiderthedualaffinequermassintegrals. Wefirstprovidealowerbound
which is sharp up to the isotropic constant of the body.
Theorem 4.1. Let K be a centered convex body of volume 1 in Rn and let 16k 6
n−1. Then,
c
(4.1) Φ˜ (K)> .
[k] L
K
Proof. BythelinearinvarianceofΦ˜ (K),wemayassumethatK isintheisotropic
[k]
position. LetF beak-dimensionalsubspaceofRn. WedenotebyE theorthogonal
subspace of F and for every φ ∈ F \{0} we define E+(φ) = {x ∈ span{E,φ} :
hx,φi>0}. K. Ball (see [1] and [19]) proved that, for every q >0, the function
!− 1
(4.2) φ7→kφk1+q+q1 Z hx,φiqdx q+1
2
K∩E+(φ)
is the gauge function of a convex body B (K,F) on F. We will make use of the
q
fact that, if K is isotropic then
L
(4.3) |K∩F⊥|1/k ’ Bk+1(K,F).
L
K
See [19] and [23] for a proof. Therefore,
! 1
Z kn
(4.4) Φ˜ (K)L ’ Lkn dν (F) .
[k] K Bk+1(K,F) n,k
Gn,k
Recall that the isotropic constant is uniformly bounded from below: we know that
L >c, where c>0 is an absolute constant. It follows that
Bk+1(K,F)
! 1
Z kn
(4.5) Φ˜ (K)L ’ Lkn dν (F) >c,
k K Bk+1(K,F) n,k
Gn,k
10