Inequalities for the surface area of projections of convex bodies

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by  Apostolos Giannopoulos

Inequalities for the surface area of projections of
convex bodies
Apostolos Giannopoulos, Alexander Koldobsky and Petros Valettas
6
1
0
2 Abstract

n We provide general inequalities that compare the surface area S(K) of a convex body K in Rn to
a the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We
J discuss the same questions for all the quermassintegrals of K. We examine separately the dependence

1 of the constants on the dimension in the case where K is in some of the classical positions or K is
2
a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by

projections and volumeby surface area.
]
G
M 1 Introduction
.
h
The starting point of this article are two inequalities of the second named author about the surface area of
t
a hyperplane projections of projection bodies. In [10] it was proved that if Z is a projection body in Rn then
m
[ (1.1) |Z|n1 ξ∈mSinn−1S(Pξ⊥(Z))6bnS(Z),

1
v where S(A) denotes the surface area of A and
0
0 (n 1)ωn−1
(1.2) b = − ,
6 n n−1
nω n
5 n
0 where ω is the volume of the Euclidean unit ball Bm in Rm. Note that b 1 and that (1.1) is sharp;
1. there is emquality if Z =Bn. Conversely,in [11] it was p2rovedthat if Z is a prnoje≃ction body in Rn which is a
2
0
dilate of a body in isotropic position, then
6
v:1 (1.3) |Z|n1 ξ∈mSanx−1S(Pξ⊥(Z))>c(logn)−2S(Z),
i
X where c>0 is an absolute constant.
Ourfirstaimistodiscusssimilarinequalitiesforthesurfaceareaofhyperplaneprojectionsofanarbitrary
r
a convex body K in Rn. In what follows, we denote by ∂ the minimal surface area parameter of K, defined
K
by
n−1
(1.4) ∂K :=min S(T(K))/T(K) n :T GL(n) .
| | ∈
n o
Itis knownthatc√n6∂ 6c′nforeveryconvexbody K inRn,wherec,c′ >0 areabsoluteconstants(see
K
Section 2 for definitions, references and backgroundinformation).
Our analogue of (1.1) is the following theorem.
Theorem 1.1. There exists an absolute constant c >0 such that, for every convex body K in Rn,
1
2b ∂ c ∂
(1.5) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))6 nnωn1K S(K)6 √1 nK S(K).
n
1
This inequality is sharp e.g. for the Euclidean unit ball. Note that c ∂ /√n 6 c√n for every convex
1 K
body K in Rn, and hence we have the general upper bound
(1.6) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))6c√nS(K).
Our method employs an estimate for the minimal volume of a hyperplane projection of K: one has
(1.7) ξ∈mSinn−1|Pξ⊥(K)|6c√n|K|nn−1
for an absolute constant c>0.
Assuming that K is in the minimal surface area position we have a converse of Theorem 1.1:
Theorem 1.2. Let K be a convex body in Rn which is in the minimal surface area position. Then,
c
(1.8) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))> √nS(K),
where c>0 is an absolute constant.
The estimate of Theorem1.2 is sharp; we provide an example in which the two quantities in (1.8) are of
the same order, using extremal (with respect to minimal hyperplane projections) bodies of minimal surface
area that were constructed in [12].
Inthe casewhere K is a projectionbody, onecansee that(1.7)holds true withc√n replacedby b (see
n
Section 3). This leads to an alternative proof of (1.1) with a weaker (by a factor of 2) constant.
Theorem 1.3. Let Z be a projection body in Rn. Then,
(1.9) |Z|n1 ξ∈mSinn−1S(Pξ⊥(Z))62bnS(Z).
Itshouldbenotedthatthereareconvexbodieswhicharenotprojectionbodiesbuttheirminimalsurface
areaparameter∂ is of the orderof√n; anexample is givenby Bn, the unit ballof ℓn. Onthe other hand,
K 1 1
there existprojectionbodies whose minimal surfacearea parameteris ofthe order ofn; anexample is given
by the cube. Thus, the estimates of Theorem 1.1 and Theorem 1.3 complement each other. In our next
result we replace min S(Pξ⊥(K)) by the expectation of S(Pξ⊥(K)) on the sphere.
Theorem 1.4. Let K be a convex body in Rn. Then,
2(n 1)ω c
(1.10) |K| ZSn−1S(Pξ⊥(K))dσ(ξ)6 −n2ωnn−1S(K)2 6 √2nS(K)2,
where c >0 is an absolute constant.
2
A consequence of Theorem 1.4 is that if K is in some of the classical positions (minimal surface area,
isotropic or John’s position, or it is symmetric and in Lo¨wner’s position) then
(1.11) |K|n1 ZSn−1S(Pξ⊥(K))dσ(ξ)6c√nS(K).
Thereasonisthat,inallthesecases,thesurfaceareaofKsatisfiesaninequalityoftheformS(K)6cnK nn−1
| |
(see Section 2 for a brief description of the classical positions of a convex body and for a proof of this last
assertion).
Passing to lower bounds, our analogue of (1.3) is the following theorem.
2
Theorem 1.5. Let K be a convex body in Rn. Then,
(1.12) S(Pξ⊥(K))dσ(ξ)>c3S(K)nn−−12,
ZSn−1
where c >0 is an absolute constant.
3
AconsequenceofTheorem1.5isthatifK isintheminimalsurfacearea,minimalmeanwidth,isotropic,
John or Lo¨wner position, then
(1.13) |K|n1 ZSn−1S(Pξ⊥(K))dσ(ξ)>cS(K),
where c>0 is an absolute constant. In particular,
(1.14) |K|n1 ξ∈mSanx−1S(Pξ⊥(K))>cS(K).
Note that (1.14) is strongerthan (1.3); moreover,for bounds of this type there is no need to assume that K
is a projection body. In fact, our proof of Theorem 1.5 shows that (1.13) continues to hold as long as
(1.15) S(K)n−11 6cK n1
| |
for an absolute constant c>0. This is a mild condition which is satisfied not only by the classical positions
but also by all reasonable positions of K.
All these inequalities are proved in Section 4. Our main tools are a result from [7] stating that
(1.16) S(Pξ⊥(K)) 6 2(n−1)S(K)
|Pξ⊥(K)| n |K|
for every convex body K in Rn and any ξ Sn−1, estimates from [9] for the volume of the projection body
∈
of a convex body in terms of its minimal surface area parameter, and Aleksandrov’s inequalities. For the
class of projection bodies, we prove and use the following sharp estimate (Lemma 3.1): if Z is a projection
body in Rn then
(1.17) ξ∈mSinn−1|Pξ⊥(Z)|6 nnbn1|Z|nn−1.
−
In Section 5 we study the same questions for the quermassintegrals V (K) = V((K,n k),(Bn,k))
n−k − 2
of a convex body K and the corresponding quermassintegrals of its hyperplane projections. We obtain the
following estimates:
(i) For every 16p6n 2 we have
−
(p+1)ω ∂ c(p+1)∂
(1.18) |K|n1 ξ∈mSinn−1Vn−1−p(Pξ⊥(K))6 nωnn−1 K Vn−p(K)6 √n K Vn−p(K)
and
(p+1)ω S(K)
(1.19) |K|n1 ZSn−1Vn−1−p(Pξ⊥(K))dσ(ξ)6 nωnn−1|K|nn−1 Vn−p(K).
(ii) If Z is a projection body in Rn then, for every 16p6n 2 we have
−
(1.20) |Z|n1 ξ∈mSinn−1Vn−1−p(Pξ⊥(Z))6(p+1)bnVn−p(Z).
3
(iii) If K is in the minimal surface area, isotropic or John’s position, or it is symmetric and in Lo¨wner’s
position, then, for every 16p6n 2 we have
−
(1.21) |K|n1 ZSn−1Vn−1−p(Pξ⊥(K))dσ(ξ)6c(p+1)√nVn−p(K).
(iv) For every 16p6n 2 we have
−
(1.22) Vn−1−p(Pξ⊥(K))dσ(ξ)> ωnn−−1−1p [Vn−p(K)]nn−−1−pp.
ZSn−1 ωnn−p
(v) If K is in the minimal surface area, isotropic or John’s position, or it is symmetric and in Lo¨wner’s
position then, for every 16p6n 2 we have
−
p
(1.23) |K|n1 ZSn−1Vn−1−p(Pξ⊥(K))dσ(ξ)> ωωn−nn1n−c−10−pn−pp Vn−p(K)>(cid:16)cn1(cid:17)2(np−p) Vn−p(K).
The proofs employ the same tools as in the surface area case. The main additional ingredient is a general-
izationof (1.16) to subspacesof arbitrarydimension andquermassintegralsofany order,provedin [5]: If K
is a convex body in Rn and 06p6k 6n, then, for every F G ,
n,k
∈
V (K) 1 V (P (K))
(1.24) n−p > k−p F .
K n−k+p P (K)
F
| | n−k | |
This inequality allows us to obtain further gener(cid:0)alizatio(cid:1)ns of the results of Section 4; we can compare the
surface area of a convex body K to the minimal, average or maximal surface area of its lower dimensional
projections P (K), F G , for any given 16k 6n 1. This is done in Section 6.
F n,k
∈ −
There are several questions that arise from this work and we hope that the reader might find them
interesting; these are stated explicitely throughout the text.
2 Notation and background
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 ballandSn−1 for the unit sphere. We denote the unit
2
ballofℓn byBn,16p6 . Inparticular,wealsowriteQ forthe cubeBn =[ 1,1]n andC = 1,1 n
p p ∞ n ∞ − n −2 2
forthecubeofvolume1. Volumeisdenotedby . Wewriteω forthevolumeofBnandσfortherotationally
invariant probability measure on Sn−1. The G|r·|assmann mannifold G of k-dime2nsional subspaces(cid:2)of Rn(cid:3)is
n,k
equipped with the Haar probability measure ν . For every 1 6 k 6 n 1 and F G we write P for
n,k n,k F
the orthogonalprojection from Rn onto F, and we set B =Bn F and−S =Sn−1∈ F. Finally, we write
F 2 ∩ F ∩
A for the homothetic image of volume 1 of a symmetric convex body A Rn, i.e. A:= A−n1A.
⊆ | |
The letters c,c′,c ,c etc. denote absolute positive constants which 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,ifK,L Rn wew≃illwriteK Lifthereexistabsoluteconstantsc ,c >0suchthatc K L c K.
1 2 1 2
⊆ ≃ ⊆ ⊆
We refer to the books [6] and [14] for basic facts from the Brunn-Minkowski theory and to the book
[1] for basic facts from asymptotic convex geometry. We also refer to [3] for more information on isotropic
convex bodies.
2.1. Convex bodies. A convex body in Rn is a compact convex subset K of Rn with non-empty interior.
WesaythatK issymmetricifx K impliesthat x K,andthatK iscenteredifitsbarycenter 1 xdx
∈ − ∈ |K| K
is at the origin. The support function of a convex body K is defined by h (y)=max x,y :x K , and
K {h i ∈ R}
the mean width of K is
(2.1) w(K)= h (θ)dσ(θ).
K
ZSn−1
4
The circumradius of K is the quantity R(K) = max x : x K i.e. the smallest R > 0 for which
2
{k k ∈ }
K RBn. If 0 int(K) then we write r(K) for the inradius of K (the largest r > 0 for which rBn K)
⊆ 2 ∈ 2 ⊆
and we define the polar body K◦ of K by
(2.2) K◦ := y Rn : x,y 61for allx K .
{ ∈ h i ∈ }
The volume radius of K is the quantity vrad(K) = (K /Bn )1/n. Integration in polar coordinates shows
| | | 2|
that if the origin is an interior point of K then the volume radius of K can be expressed as
1/n
(2.3) vrad(K)= θ −ndσ(θ) ,
(cid:18)ZSn−1k kK (cid:19)
where θ =min t>0:θ tK . We also define
K
k k { ∈ }
(2.4) M(K)= θ dσ(θ).
K
ZSn−1k k
2.2. Mixed volumes. FromMinkowski’sfundamentaltheoremweknowthatifK ,...,K arenon-empty,
1 m
compactconvexsubsetsofRn,thenthevolumeoft K + +t K isahomogeneouspolynomialofdegree
1 1 m m
···
n in t >0. That is,
i
(2.5) t K + +t K = V(K ,...,K )t t ,
| 1 1 ··· m m| i1 in i1··· in
16i1,X...,in6m
where the coefficients V(K ,...,K ) are chosen to be invariant under permutations of their arguments.
i1 in
The coefficientV(K ,...,K ) is the mixed volume ofK ,...,K . Inparticular,if K andC are two convex
1 n 1 n
bodies in Rn then the function K+tC is a polynomial in t [0, ):
| | ∈ ∞
n
n
(2.6) K+tC = V (K,C)tj,
n−j
| | j
j=0(cid:18) (cid:19)
X
where V (K,C)=V((K,n j),(C,j)) is the j-th mixed volume of K and C (we use the notation (C,j)
n−j
−
for C,...,C j-times). If C =Bn then we set V (K):=V (K,Bn)=V((K,n j),(Bn,j)); this is the
2 n−j n−j 2 − 2
j-th quermassintegralof K. Note that
1 K+tC K
(2.7) V (K,C)= lim | |−| |,
n−1
nt→0+ t
and by the Brunn-Minkowskiinequality we see that
(2.8) Vn−1(K,C)> K nn−1 C 1/n
| | | |
for all K and C (this is Minkowski’s first inequality). The mixed volume V (K,C) can be expressed as
n−1
1
(2.9) V (K,C)= h (θ)dσ (θ),
n−1 C K
nZSn−1
where σ is the surface area measure of K. In particular, the surface area of K satisfies
K
(2.10) S(K)=nV (K).
n−1
We will also use the Aleksandrov inequalities: if K is a convex body in Rn then the sequence
1
1 k
(2.11) Q (K)= P (K) dν (F)
k F n,k
ωk ZGn,k| | !
5
is decreasing in k. This is a consequence of the Aleksandrov-Fenchel inequality (see [4] and [14]). In
particular, for every 16k 6n 1 we have
−
K n1 1 k1
(2.12) | | 6 P (K) dν (F) 6w(K).
F n,k
(cid:18)ωn (cid:19) ωk ZGn,k| | !
2.3. Classical positions. Let K be a centered convex body in Rn. We introduce the classical positions of
K that we are going to discuss; we set the notation and provide some backgroundinformation.
Minimal surface area position. We say that K has minimal surface area if S(K) 6 S(T(K)) for every
T SL(n). Petty ([13], see also [9]) proved that K has minimal surface area if and only if the measure σ
K
∈
satisfies the isotropic condition
(2.13) S(K)=n ξ,θ 2dσ (θ)
K
ZSn−1h i
for every ξ Sn−1. From the isoperimetric inequality we know that S(K) > nωnn1 K nn−1. The reverse
∈ | |
isoperimetricinequality ofK.Ball[2] implies thatif K hasminimal surfaceareaandvolume1 then S(K)6
S(C ) = 2n in the symmetric case and S(K) 6 S(∆ ) 6 c n in the not necessarily symmetric case, where
n n 0
∆ is a regular simplex of volume 1 in Rn and c >0 is an absolute constant.
n 0
Minimal mean width position. We say that K is in minimal mean width position if w(K) 6 w(T(K)) for
every T SL(n). It was proved in [8] that K has minimal mean width if and only if
∈
(2.14) w(K)=n ξ,θ 2h (θ)dσ(θ)
K
ZSn−1h i
for every ξ Sn−1. From results of Figiel-Tomczak,Lewis and Pisier (see [1, Chapter 6]) it follows that if a
symmetric c∈onvex body K in Rn has minimal mean width then
(2.15) M(K)w(K)6c log(d +1)
1 K
where d := d(K,Bn) is the Banach-Mazur distance from K to Bn and c > 0 is an absolute constant. If
K 2 2 1
we assume that K = 1 then w(K) 6 c√nlog(d +1). Then, a simple argument shows that any convex
K
body of volume 1| in| Rn that has minimal mean width satisfies a similar bound: w(K)6c√nlogn.
Isotropic position. For every centered convex body K of volume 1 in Rn and any q >1 we define
1/q
(2.16) I (K)= x qdx .
q k k2
(cid:18)ZK (cid:19)
We say that K is in the isotropic position if I (K) 6 I (T(K)) for every T SL(n). This is equivalent to
2 2
∈
the existence of a constant L >0 such that
K
(2.17) x,ξ 2dx=L2
h i K
ZK
for every ξ Sn−1. It is known that if K is centered then
∈
(2.18) x,ξ 2dx K ξ⊥ −2
h i ≃| ∩ |
ZK
foreveryξ Sn−1. Therefore,ifK isisotropicweseethatallhyperplanesectionsK ξ⊥ ofK havevolume
equal (up to∈an absolute constant) to L−1. ∩
K
John and Lo¨wner position. We say that a convex body K is in John’s position if the ellipsoid of maximal
volume inscribed in K is a multiple of the Euclidean unit ball Bn. We say that a convex body K is in
2
6
Lo¨wner’s position if the ellipsoid of minimal volume containing K is a multiple of the Euclidean unit ball
Bn. OnecancheckthatthisholdstrueifandonlyifK◦ isinJohn’sposition. Thevolumeratioofacentered
2
convex body K in Rn is the quantity
1
K n
(2.19) vr(K)=inf | | : is an ellipsoid and K .
((cid:18)|E|(cid:19) E E ⊆ )
TheoutervolumeratioofacenteredconvexbodyK inRn isthe quantityovr(K)=vr(K◦). K.Ballproved
in[2]thatifK isinJohn’spositionthenvr(K)6vr(C ) √ninthesymmetriccaseandvr(K)6vr(∆ )
n n
≃ ≃
√n in the not necessarilysymmetric case;in fact, the reverseisoperimetric inequality follows from this fact.
2.4. Surface area and inradius. Let K be a centered convex body in Rn. Recall that the inradius r(K)
of K is the largest r >0 for which rBn K. Using the monotonicity of mixed volumes we may write
2 ⊆
1
(2.20) S(K)=nV (K,Bn)6nV K, K .
n−1 2 n−1 r(K)
(cid:16) (cid:17)
Since the mixed volumes are homogeneous with respect to each of their arguments and V(K,...,K)= K ,
| |
we have the following general estimate for the surface area S(K) of K.
Lemma 2.1. Let K be a convex body in Rn with 0 int(K). Then,
∈
nK
(2.21) S(K)6 | |.
r(K)
Using Lemma 2.1 we obtain upper bounds for the surface area of a body which is in isotropic, John’s or
Lo¨wner’s position.
Proposition 2.2. Let K be a centered convex body of volume 1 in Rn.
(i) If K is isotropic then S(K)6cn/L 6c′n, where c,c′ >0 are absolute constants.
K
(ii) If K is in minimal surface area position or in John’s position then S(K) 6 cn, where c > 0 is an
absolute constant.
(iii) If K is symmetric and in Lo¨wner’s position then S(K)6cn, where c>0 is an absolute constant.
(iv) If K is symmetric and in the minimal mean width position then S(K) 6 cnlogn, where c > 0 is an
absolute constant.
Proof. The inclusion L Bn K for an isotropic symmetric convex body K in Rn is clear since
K 2 ⊆
hK(u)= ,u L∞(K) > ,u L2(K) =LK
kh· ik kh· ik
foreveryu Sn−1. Thisshowsthatr(K)>L inthiscase. IfK iscenteredbutnotnecessarilysymmetric,
K
then we stil∈l have h (u)>cL : to see this, we use the fact that cmax K (tθ+θ⊥) :t R 6 K θ⊥
K K
{| ∩ | ∈ } | ∩ |
(see [3, Chapter 2]) and then write
∞
L−1h (u)>c h (u)K θ⊥ >c K (tθ+θ⊥) dt
K K 1 K | ∩ | 2 | ∩ |
Z0
=c x K : x,θ >0 >c ,
2 3
|{ ∈ h i }|
where c > 0 is an absolute constant (the last inequality follows from Gru¨nbaum’s lemma, see [3, Chapter
3
2]). To conclude the proof we recall that L >c for any convex body K in Rn.
K
Assume that K is in John’s position. Then, using the volume ratio estimate we see that
1
√n K n
(2.22) | | =vr(K)6c√n,
r(K) ≃ r(K)Bn
(cid:18)| 2|(cid:19)
7
whichimpliesthatr(K)>c,andhenceS(K)6c−1n. ItfollowsthatifK isinminimalsurfaceareaposition
we also have S(K)6c−1n.
Next, assume that K is symmetric and in Lo¨wner’s position; this time we use the fact that R(K) 6
√nr(K) by John’s theorem, and then
(2.23) 1= K 1/n 6 R(K)Bn 1/n 6cR(K)/√n6cr(K).
| | | 2|
Finally, if K is symmetric and in the minimal mean width position we can use the direct estimate
(2.24) R(K◦)6c√nw(K◦)=c√nM(K)6c′logn
which is a consequence of (2.15) andof the factthat w(K)>c√n by Urysohn’s inequality. This shows that
r(K)=1/R(K◦)>c/logn, and (iv) follows. ✷
Note. The example of the cube C shows that the bounds (i), (ii) and (iii) of Proposition 2.2 are sharp up
n
to an absolute constant.
3 Projections of projection bodies
Azonoidis the limitofMinkowskisumsoflinesegmentsinthe Hausdorffmetric. Equivalently,asymmetric
convex body Z is a zonoid if and only if its polar body is the unit ball of an n-dimensional subspace of an
L -space; i.e. if there exists a positive measure µ (the supporting measure of Z) on Sn−1 such that
1
1
(3.1) hZ(x)= x Z◦ = x,y dµ(y).
k k 2ZSn−1|h i|
Theclassoforigin-symmetriczonoidscoincideswiththeclassofprojectionbodies. Recallthattheprojection
body ΠK of a convex body K is the symmetric convex body whose support function is defined by
(3.2) hΠK(ξ)=|Pξ⊥(K)|, ξ ∈Sn−1.
From Cauchy’s formula
1
(3.3) |Pξ⊥(K)|= 2ZSn−1|hξ,θi|dσK(θ),
whereσ isthesurfaceareameasureofK,weseethattheprojectionbodyofK isazonoidwhosesupporting
K
measure is σ . Minkowski’s existence theorem implies that, conversely, every origin-symmetric zonoid is
K
the projection body of some symmetric convex body in Rn. Moreover, if we denote by the class of
n
C
origin-symmetric convex bodies and by the class of origin-symmetric zonoids, Aleksandrov’s uniqueness
n
Z
theorem shows that the Minkowski map Π : with K ΠK, is injective. Note also that is
n n n
C → Z 7→ Z
invariant under invertible linear transformations and closed in the Hausdorff metric.
Let K be a convex body of volume 1 in Rn. Then, the volume of ΠK and of its polar body Π∗K satisfy
the bounds (see [9])
∂ n ω ∂ n
(3.4) K 6 ΠK 6ω n−1 K
n
n | | nω
(cid:18) (cid:19) (cid:18) n (cid:19)
and
nω n 4nnn
(3.5) ω n 6 Π∗K 6 .
n ω ∂ | | n!∂n
(cid:18) n−1 K(cid:19) K
All these inequalities are sharp as one can see from the examples of the ball and the cube.
Our next lemma provides an estimate for the volume of the minimal hyperplane projection of a zonoid.
8
Lemma 3.1. Let Z be a zonoid in Rn. Then,
(3.6) ξ∈mSinn−1|Pξ⊥(Z)|6 nnbn1|Z|nn−1.
−
Proof. We write Z =ΠK for some convex body K. Recall the volume formula for zonoids
1 1
(3.7) |Z|=Vn−1(Z,ΠK)=Vn−1(K,ΠZ)= nZSn−1hΠZ(ξ)dσK(ξ)= nZSn−1|Pξ⊥(Z)|dσK(ξ).
Therefore,
S(K)
(3.8) |Z|> n ξ∈mSinn−1|Pξ⊥(Z)|.
On the other hand, by (3.4) we have
(3.9) |Z|n1 =|ΠK|n1 6 ωn−n1−∂1K|K|nn−1 6 bnnS(K1),
nωnn −
where we used the definition (1.2) of the constant bn and the fact that S(K)>∂K K nn−1 by the definition
| |
(1.4) of the minimal surface area parameter ∂ . Then,
K
(3.10) S(nK)ξ∈mSinn−1|Pξ⊥(Z)|6|Z|n1|Z|nn−1 6 bnnS(K1)|Z|n1,
−
and the result follows. ✷
Sinceeveryprojectionofazonoidisazonoid,asimpleinductionargumentleadstothefollowinggeneral
result.
Theorem 3.2. Let Z be a zonoid in Rn. Then, for every 16k 6n 1 we have
−
nbn−k
(3.11) min PF(Z) 6 n Z nk.
F∈Gn,k| | k | |
Question 3.3. The exampleofthe ballshowsthat Lemma3.1is sharp. Itwouldbe interestingtoestablish
the precise constant in Theorem 3.2.
4 Surface area of hyperplane projections
Our generalization of (1.1) is in terms of the minimal surface area parameter ∂ of K.
K
Theorem 4.1. Let K be a convex body in Rn. Then,
2b ∂ c ∂
(4.1) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))6 nnωn1K S(K)6 √1 nK S(K),
n
where c > 0 is an absolute constant and ∂ is the minimal surface area parameter of K. Therefore, in
1 K
general we have that
(4.2) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))6c√nS(K).
The main ingredient in the proof is the next result from [7].
9
Lemma 4.2. If K is a convex body in Rn then
(4.3) S(Pξ⊥(K)) 6 2(n−1)S(K)
|Pξ⊥(K)| n |K|
for every ξ Sn−1.
∈
Proof of Theorem 4.1. From (4.3) we have
2(n 1)
(4.4) |K|S(Pξ⊥(K))6 n− S(K)|Pξ⊥(K)|
for every ξ Sn−1. Therefore,
∈
2(n 1)
(4.5) |K| ξ∈mSinn−1S(Pξ⊥(K))6 n− S(K) ξ∈mSinn−1|Pξ⊥(K)|.
Next, we observe that
(4.6) ξ∈mSinn−1|Pξ⊥(K)|=ξ∈mSinn−1hΠK(ξ)=r(ΠK).
Since
(4.7) |ΠK|n1 6 ωn−n1−∂1K |K|nn−1
nω n
n
we get
(4.8) r(ΠK)6vrad(ΠK)6 ωn−1∂K K nn−1.
nω | |
n
Going back to (4.5) we see that
(4.9) |K| ξ∈mSinn−1S(Pξ⊥(K))6 2(n−n12)ωωnn−1∂K S(K)|K|nn−1,
and this proves (4.1). ✷
Question 4.3. It would be interesting to decide whether there exist convex bodies K such that
(4.10) |K|n1 ξ∈mSinn−1S(Pξ⊥(K))>c√nS(K).
This would show that Theorem 4.1 is asymptotically sharp. Note that in the case of the Euclidean ball one
has
(4.11) |B2n|n1 ξ∈mSinn−1S(Pξ⊥(B2n))>cS(B2n).
We can prove an inequality which is reverse to (4.1) for any convex body K which is in the minimal
surface area position, using the following fact (see Theorem 1.2 in [7]): for any convex body K in Rn and
any ξ Sn−1 one has
∈
(4.12) Vn−2(K) 6 Vn−2(Pξ⊥(K)).
2Vn−1(K) |Pξ⊥(K)|
Note that
(4.13) Vn−2(K)=ωn[Qn−2(K)]n−2 >ωn[Qn−1(K)]n−2 =ωnn−11[Vn−1(K)]nn−−12,
10