Stability of the Blaschke-Santaló and the affine isoperimetric inequality

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by  Károly J. Böröczky

Stability of the Blaschke-Santalo´ and the affine
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isoperimetric inequality
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a Ka´roly J. Bo¨ro¨czky
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January 21, 2009
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Dedicated toEndreMakaion theoccasionofhissixtiethbirthday
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a Abstract
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A stability version of the Blaschke-Santalo´ inequality and the affine
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isoperimetricinequalityforconvexbodiesofdimensionn 3isproved. The
1 ≥
v first step is the reduction to the case when the convex body is o-symmetric
0 and has axial rotational symmetry. This step works for related inequalities
4
compatiblewithSteinersymmetrization. Secondly,fortheseconvexbodies,
3
3 a stability version of the characterization of ellipsoids by the fact that each
. hyperplane sectioniscentrally symmetricisestablished.
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9 2000MathematicsSubjectClassification: 52A40
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:
v
i 1 Introduction
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r
a
Stability versions of geometric inequalities have been investigatedsince the days
of H. Minkowski, see the beautiful survey of H. Groemer [19], or K.J. Bo¨ro¨czky
[10]for somemore recent results. Here we provestabilityversionsof two classi-
cal inequalities originating from the beginningof the 20th century, the Blaschke-
Santalo´ inequality and the affine isoperimetric inequality. For all the basic affine
invariantnotions,consultthethoroughmonographofK.Leichtweiß[32], andfor
notionsofconvexityingeneral, seeP.M. Gruber[23]and R. Schneider[50].
Supportedby OTKA grants068398and 049301,and by the EU Marie Curie TOK projects
∗
DiscConvGeoandBudAlgGeo.
1
WewriteotodenotetheoriginofRn, , todenotethestandardscalarprod-
h· ·i
uct, and V( ) to denote volume. Let Bn be the unit Euclidean ball with volume
·
k =V(Bn), and let Sn 1 =¶ Bn. A convexbody K in Rn isa compact convexset
n −
with non–empty interior. If z intK, then the polar of K with respect to z is the
∈
convexbody
Kz = x Rn : x z,y z 1 foranyy K .
{ ∈ h − − i≤ ∈ }
Itiseasytoseethat(Kz)z=K. AccordingtoL.A.Santalo´ [49](seealsoM.Meyer
and A. Pajor [38]), there exists a unique z intK minimizingthe volumeproduct
∈
V(K)V(Kz), which is called the Santalo´ point of K. In this case z is the centroid
of Kz. The celebrated Blaschke-Santalo´ inequality states that if z is the Santalo´
point(orcentroid)ofK, then
V(K)V(Kz) k 2, (1)
n
≤
with equality if and only if K is an ellipsoid. The inequality was proved by W.
Blaschke [7] for n 3, and by L.A. Santalo´ [49] for all n. The case of equality
≤
was characterized by J. Saint-Raymond [48] among o-symmetric convex bodies,
andbyC.M.Petty[44]amongallconvexbodies(seealsoM.MeyerandA.Pajor
[38], D.Hug [24], andM. Meyerand S. Reisner[39]forsimplerproofs).
Ourmaintaskistoprovideastabilityversionofthisinequality. Anaturaltool
istheBanach-Mazurdistanced (K,M)oftheconvexbodiesK andM,whichis
BM
defined by
d (K,M)=min l 1: K x F (M y) l (K x) for F GL(n),x,y Rn .
BM
{ ≥ − ⊂ − ⊂ − ∈ ∈ }
In particular, if K and M are o-symmetric, then x = y = o can be assumed. It
follows from a theorem of F. John [25] that d (K,Bn) n for any convex body
BM
K inRn (see alsoK.M.Ball [4]). ≤
Theorem 1.1 IfK is aconvex bodyin Rn, n 3,with Santalo´ pointz,and
≥
V(K)V(Kz)>(1 e )k 2 fore (0,1),
− n ∈ 2
thenforsomeg >0dependingonlyonn, we have
d BM(K,Bn)<1+ge 61n loge 16.
| |
2
TakingK to betheconvexbodyresultingfrom Bn by cuttingofftwo opposite
caps ofvolumee showsthat the exponent1/(6n) cannot be replaced by anything
larger than 2/(n+1). Therefore the exponent of e is of the correct order. Since
1/(6n)ismostprobablynottheoptimalexponentofe inTheorem1.1,noattempt
wasmadetofindanexplicitg inTheorem1.1. Inprinciple,thiswouldbepossible
followingtheargumentinthispaperiftheexponent1/(6n)isreplacedby1/(6n+
6)(seethediscussionafter(20)). WenotethatastabilityversionoftheBlaschke-
Santalo´ inequality in the planar case is proved in K.J. Bo¨ro¨czky, E. Makai, M.
Meyer,S. Reisner [11], usingaquitedifferent method.
The literature about the Blaschke-Santalo´ inequality is so extensive that only
justasmallportioncanbediscussedhere. ThePhDthesisofK.M.Ball[3]started
off the quest for suitable functional versions. This point of view is for example
pursued in M. Fradelizi and M. Meyer [15] and S. Artstein, B. Klartag and V. D.
Milman [2]. Stability questions on a related problem are discussed in M. Meyer
and E.Werner [40].
We note that the minimum of the volume product V(K)V(Kz) is not known
for convex bodies K in Rn and z K for n 3. According to the well-known
∈ ≥
conjectureofK. Mahler[36], thevolumeproduct is minimizedby simplices,and
amongo-symmetricconvexbodiesbycubes. Theplanarcasewasactuallyproved
in[36],andsimplerargumentsareprovidedbyM.Meyer[37]andM.Meyerand
S. Reisner [39]. For n 3, the Mahlerconjecture for o-symmetricconvex bodies
≥
hasbeenbeenverifiedamongunconditionalbodiesbyJ.Saint-Raymond[48](see
also S. Reisner [46]), and among zonoids by S. Reisner [45] (see also Y. Gordon,
M. Meyer and S. Reisner [17]). The best lower bound for the volume product of
an o-symmetricconvexbodyK inRn is
V(K)V(Ko)>2 nk 2, (2)
− n
due to G. Kuperberg [29]. With a non-explicit exponential factor instead of 2 n,
−
itwas provedby J. Bourgainand V.D. Milman[12].
The Mahler conjecture for general convex bodies was verified by M. Meyer
and S. Reisner [39] among polytopes of at most n+3 vertices. In a yet unpub-
lished revision of [29], G. Kuperberg also showed, based on (2) and the Rogers-
Shephard inequality[47], that ifz intK foraconvexbodyK inRn, then
∈
V(K)V(Kz)>4 nk 2. (3)
− n
It was probably W. Blaschke who first noticed that the Blaschke-Santalo´ in-
equalityisequivalenttotheaffineisoperimetricinequality. Thisandotherequiva-
lentformulationsarediscussedindepthinE.Lutwak[35]andK.Leichtweiß[32],
3
Section 2. To definetheaffine surface area ofaconvexbodyK in Rn , wealways
consider its boundary endowed with the (n 1)-dimensional Hausdorff measure.
−
According to Alexandrov’s theorem (see P.M. Gruber [23], p. 74), ¶ K is twice
differentiable in a generalized sense at almost every point, hence the generalized
Gauß-Kroneckercurvaturek (x)canbedefinedatthesex ¶ K (seeK.Leichtweiß
∈
[32], Section1.2). Theaffinesurfacearea ofK isdefined by
W (K)= k (x)n+11dx.
Z¶ K
If¶ K isC2,thenthisdefinitionisduetoW.Blaschke[6]. Sincethenvariousequiv-
alent definitions were given for general convex bodies (including the above one)
by K. Leichtweiß [31], C. Schu¨tt and E. Werner [52] and E. Lutwak [34], which
wereshowntobeequivalentbyC.Schu¨tt[51],andG.DolzmannandD.Hug[14]
(seeK.Leichtweiß[32],Section2). Theaffinesurfaceareaisavaluationinvariant
under volumepreserving affine transformations, and it is upper semi-continuous.
Thesepropertiesarecharacteristic,asanyuppersemi-continuousvaluationonthe
space of convex bodies which is invariant under volume preserving affine trans-
formations is a linear combination of affine surface area, volume, and the Euler
characteristicbyM.LudwigandM.Reitzner[33]. Wenotethataffinesurfacearea
comes up e.g. in polytopal approximation (see P.M. Gruber [23], Section 11.2),
inlimitshapeoflatticepolygons(seeI. Ba´ra´ny [5]), and manyotherapplications
(seeK. Leichtweiß[32], Section 2).
Theaffineisoperimetricinequalitystates that
W (K)n+1 k 2nn+1V(K)n 1, (4)
n −
≤
with equality if and only if K is an ellipsoid. The inequality itself is due to W.
Blaschke [6], whose proof in R3 for convex bodies with C2 boundaries readily
extends to general dimension and to general convex bodies. W. Blaschke charac-
terized the equality case among convex bodies with C2 boundary, and this char-
acterization was extended to all convex bodies by C.M. Petty [44]. We note that
W. Blaschke and L.A. Santalo´ deduced the Blaschke-Santalo´ inequality from the
affineisoperimetricinequality. Here wetakeareversepath.
Aninequalityonp. 59ofE.Lutwak[34](seealsoLemma3.7inD.Hug[24],
or(1106)in K.Leichtweiß[32]) saysthatifz intK,then
∈
W (K)n+1 nn+1V(K)nV(Kz).
≤
ThereforeTheorem 1.1yields
4
Theorem 1.2 IfK is aconvex bodyin Rn, n 3,and
≥
W (K)n+1 >(1 e )k 2nn+1V(K)n 1 for e (0,1),
− n − ∈ 2
thenforsomeg >0dependingonlyonn, we have
d BM(K,Bn)<1+ge 61n loge 16.
| |
ForconvexbodiesK and M, wewriteV (K,M)todenotethemixedvolume
1
V(K+tM) V(K)
V (K,M)=lim −
1
t 0 n t
→ ·
(seeT.BonnesenandW.Fenchel[9],Section29,orP.M.Gruber[23],Section6).
It satisfiesV (K,K)=V(K). We write K n to denote the family of convex bodies
1 o
whosecentroid iso. C.M.Petty [42]defined thegeominimalsurfacearea by
G(K)=k −n1/nninf V1(K,Mo)V(M)n1 :M Kon .
{ ∈ }
It isalso invariantundervolumepreservingaffine transformations. PositioningK
in a way such that o is the Santalo´ point of K and taking M = Ko, yields the so
called geominimalsurfacearea inequalityofC.M. Petty [43]
G(K) k n1/nnV(K)n−n1, (5)
≤
withequalityifandonlyifK isanellipsoid. FromTheorem1.1wedirectlyobtain
Theorem 1.3 IfK is aconvex bodyin Rn, n 3,and
≥
G(K)>(1−e )k n1/nnV(K)n−n1 fore ∈(0,21),
thenforsomeg >0dependingonlyonn, we have
d BM(K,Bn)<1+ge 61n loge 16.
| |
One of our main tools is to reduce the proof of Theorem 1.1 to o-symmetric
convexbodieswith axialrotationalsymmetry.
Theorem 1.4 For any convex body K in Rn, n 2, with d (K,Bn)>1+e for
BM
e >0, there exists an o-symmetric convex body≥C with axial rotational symmetry
andaconstantg >0dependingonnsuchthatd (C,Bn)>1+ge 2 andC results
BM
from K as a limit of subsequent Steiner symmetrizations and affine transforma-
tions.
5
Remark: IfK iso-symmetric,then1+ge 2 canbereplacedby1+ge . Inparticu-
lar,ifK iso-symmetric,thenwhereverthefactor1/6occursinTheorems1.1,1.2
and 1.3, itcan bereplaced by1/3.
Theorem 1.4 shows that it is possibleto use Steiner-symmetrizationto obtain
a convex body that is highly symmetric but still far from being an ellipsoid. On
the other hand, B. Klartag [27] proved that any convex body K in Rn gets e close
tosomeballaftersuitablechosencn4 loge 2 Steinersymmetrizationswherec>0
| |
isan absoluteconstant.
After discussing the basic tools such as the isotropic position of convex bod-
ies and Steiner symmetrization in Section 2, we prove Theorem 1.4 in Section 3.
A stability version of the False Centre theorem in a special case is presented in
Section 4, which combined with Theorem 1.4 leads to the proof of Theorem 1.1
in Section 5. For stability versions of some other classical geometric characteri-
zationsofellipsoids,see, e.g.,H. Groemer[20]and P.M.Gruber[22].
2 Some tools
2.1 Isotropic position
In this paper, we use the term isotropic positionin a weak sense. More precisely,
we say that a convex body K in Rn is in weak isotropic position if its centroid is
o,and u,x 2dxisindependentofu Sn 1. In particular,in thiscase
K −
h i ∈
R
u,x 2dx=L2V(K)n+n2
ZKh i K
for any u Sn 1 (see, e.g., A.A. Giannopoulos and V.D. Milman [16]), and the
−
∈
Legendreellipsoid(theellipsoidofinertia)isaball. ForanyconvexbodyC there
is a volume preserving affine transformation T such that TC is in weak isotropic
position. Intheliterature,twodiffferentnormalizationsareused. EitherV(K)=1
(see, e.g., A.A. Giannopoulos and V.D. Milman [16]), or v 2 = v,x 2dx for
K
any v Rn (see, e.g., R. Kannan, L. Lova´sz and M. SimkonkovitsR[2h6]).i In this
∈
paper,ifK isinweak isotropicposition,thenwecompareittoballs,thereforewe
frequentlyassumeV(K)=k .
n
It is known that L is minimized by ellipsoids (see F. John [25] or A.A. Gi-
K
annopoulosandV.D.Milman[16]). ItfollowsbyGy. Sonnevend[53](seealsoR.
Kannan,L.Lova´szandM.Simonovits[26])thatifK isinweakisotropicposition,
6
then
2
K Ln+2V(K)n1 n(n+2)Bn.
⊂ K
Now LK c0√4 n for some absolute conpstant c0 according to B. Klartag [28].
≤
Therefore, ifV(K)=k andK is inweak isotropicposition,then
n
K c√nBn (6)
⊂
forsomeabsoluteconstantc 1.
≥
Forpropertiesofo-symmetricconvexbodiesinisotropicposition,seethedis-
cussioninV.D. Milmanand A. Pajor[41].
2.2 Steiner symmetrization
Given a convex body K in Rn and a hyperplaneH, for any l orthogonal to H and
intersectingK,translatel K alongl inawaysuchthatthemidpointoftheimage
∩
lies in H. The union of these images is the Steiner symmetrial K of K with
H
respect to H. It followsthat K is convex,V(K )=V(K), and, if thecentroid of
H H
K liesinH, thenit coincideswiththecentroidofK .
H
We write to denote the (n 1)-dimensionalLebesgue measure, where the
|·| −
measure of the empty set is defined to be zero. For u Sn 1 and t R, let u
− ⊥
∈ ∈
denote the linear (n 1)-space orthogonal to u, let h (u)=max u,x be the
K x K
− ∈ h i
supportfunctionofK, and let
K(u,t)=K (tu+u ).
⊥
∩
IfMisacompactconvexsetofdimensionn 1,thentheclassicalBrunn-Minkowski
−
inequality (see, e.g., T. Bonnesen and W. Fenchel [9], p. 94, P.M. Gruber [23],
Section8.1, orthemonographR. Schneider[50], whichissolelydedicated tothe
Brunn-Minkowskitheory)yields
1(M M) M . (7)
|2 − |≥| |
K.M. Ball proved in his PhD thesis [3] that Steiner symmetrization through
u for u Sn 1 increasesV(Ko) if K is o-symmetric. The basis of his argument
⊥ −
∈
istheobservationthatforK =K , wehave
u
⊥
1(Ko(u,t) Ko(u,t)) Ko(u,t) tu (8)
2 e − ⊂ −
(see also M. Meyer and A. Pajor [38]). Here the (n 1)-measure of the left hand
e −
side is at least Ko(u,t) according to the Brunn-Minkowski inequality, hence the
| |
7
Fubini Theorem yieldsV(Ko) V(Ko). K.M. Ball’s result was further exploited
≥
byM.MeyerandA.Pajor[38]. Theideasandstatementsin[38]yieldthefollow-
ing. e
Lemma 2.1 (Meyer,Pajor) Let K be a convex bodyin Rn, and let H be a hyper-
plane. Ifzandz denotetheSantalo´ pointsofK andK ,respectively,thenz H,
′ H ′
∈
andV(Kz) V((KH)z′).
≤
ThisstatementismoreexplicitinTheorem1ofM.MeyerandS.Reisner[39]
(seetheproofofTheorem 13in [39]).
3 Proof of Theorem 1.4
ThefollowinglemmaisthebasisoftheproofofTheorem1.4.
Lemma 3.1 Let K be a convex body in Rn. If d (K,Bn)>1+e for e >0, then
BM
there exists a convex body C with axial rotational symmetry that results from K
as a limit of subsequent Steiner symmetrizations and affine transformations, and
satisfies d (C,Bn) > 1+ge , where g > 0 depends only on n. Moreover if K is
BM
o-symmetric,then soisC.
Proof: WemayassumethatV(K)=k andK isinweakisotropicposition. Using
n
thec 1 from (6), weclaimthatthereexistssomeu Sn 1 suchthat
−
≥ ∈
(i) eitherh (u) 1+ e andV(K Bn) ge˜ forg˜ = 1 u,x 2dx,
K ≥ 4 \ ≤ 4c2n Bnh i
R
(ii) orh (u) 1 g˜ e .
K ≤ −nk n
To prove this statement, let h attain its maximum on Sn 1 at v Sn 1, and its
K − −
minimumat w Sn 1. If h (w) 1 e , then u=w works,thus w∈emay assume
h (w) 1 e ∈. Sin−ce d K(K,B≤n) >−14+e , it follows that h (v) 1+ e . Now
ifKV(K≥Bn)− 4ge˜ , then wBMe are done again, hence we may aKssum≥e V(B4n K) =
V(K B\n) ge≤˜ . We conclude that h (w) 1 g˜ e , which completes the\proof
\ ≥ K ≤ − nk n
of(i)and(ii).
LetCbetheimageofKafterapplyingfirstSchwarzrounding(seeP.M.Gruber
[23], p. 178) in the direction of u, and secondly the linear transformation that
dilates by the factor hK(u)−1 in the direction of u, and by the factor hK(u)n−11
orthogonalto u. Since Schwarz rounding can be obtainedas thelimit ofrepeated
8
applications of Steiner symmetrizations through hyperplanes containing the line
Ru,wehaveV(C)=V(K)andoisthecentroidofC (seeSection2.2). Thelinear
transformationfollowingtheSchwarz roundingensures thatu ¶ C.
Let h=hK(u)and h˜ =hK( u). In thecase of(ii),LK LB∈n yields
− ≥
1 h˜/h
u,x 2dx = r2 C(u,r) dr+ r2 C( u,r) dr
ZCh i Z0 | | Z0 | − |
1 h˜/h
= r2h K(u,hr) dr+ r2h K( u,hr) dr
Z0 | | Z0 | − |
1 h h˜
= s2 K(u,s) ds+ s2 K(u,s) ds
h2 Z0 | | Z0 | | !
1
= u,x 2dx
h2ZKh i
1 n+2
= L2k n
h2 K n
1
u,x 2dx
≥ h2ZBnh i
> (1+ g˜ e ) u,x 2dx. (9)
nk n ZBnh i
In thecaseof(i), wehaveK c√nBn according to(6). It followsthat
⊂
1
u,x 2dx = u,x 2dx
ZCh i h2ZKh i
1
< u,x 2dx+c2nV(K Bn)
h2 ZBnh i \
(cid:18) (cid:19)
e
1+
4 u,x 2dx
≤ h2 ZBnh i
< (1 e ) u,x 2dx. (10)
−8 ZBnh i
Let d (C,Bn) =1+h , where we may assume that h (0,1). Since C has
BM
axialrotationalsymmetryaroundRu,andoisthecentroido∈fC,thereexistg >0
1
depending only on n, and an o-symmetric ellipsoid E with axial rotational sym-
metryaroundRu suchthatE C (1+g h )E. It followsbyV(C)=V(Bn) and
1
⊂ ⊂
u ¶ C thatthereexistsag >0 dependingonlyonn such that
2
∈
(1+g h ) 1Bn C (1+g h )Bn.
2 − 2
⊂ ⊂
9
Therefore, we conclude Lemma 3.1 by (10) in the case of (i), and by (9) in the
caseof(ii). 2
Let us write W(M) to denote the mean width of a planar compact convex
set M. In particular p W(M) is the perimeter of M. Writing R(M) and r(M) to
denote the circum- and the inradius of M, and A(M) to denote the area of M, the
Bonnesen inequality(appearing first in W. Blaschke [8], see H. Groemer [19] for
morereferences) states
W(M)2 4A(M) (R(M) r(M))2. (11)
p
− ≥ −
To prove Theorem 1.4 for convex bodies in Rn, we need the following state-
ment.
Proposition3.2 IfM isaplanarcompactconvexsetinR2 withanaxisofsymme-
try satisfying d (K,B2)>1+e for e (0,1), then there exist orthogonal lines
BM
l andl suchthatd ((K ) ,B2)>1∈+ce 2 for c =0.001.
1 2 BM l1 l2 ′ ′
Proof: Let l be the line of symmetry of K. We may assume that A(K)= p , and
thatl intersectsK inasegmentoflength2whosemidpointiso.
FirstwetrySteinersymmetrizationthroughl,andthelinel thatisorthogonal
′
tol througho. If d ((K ) ,B2)>1+ce 2, then wearedone. Otherwisethereis
BM l l ′
′
an ellipseE whoseprincipalaxis arecontained inl andl suchthat
′
E (K ) (1+ce 2)E.
l l ′
⊂ ′ ⊂
Wededucethat
(1+ce 2) 3B2 (K ) . (12)
′ − l l
⊂ ′
Since d (K,B2) > 1+e , it follows that R(K) r(K) e /2. Therefore, the
BM
− ≥
Bonnesen inequality(11)yields
1
e 2 2
W(K) 2 1+ .
≥ · 16
(cid:18) (cid:19)
In particular,ifthedistanceofx ,x ¶ K isthediameterofK, then
1 2
∈
x x >2 (1+ce 2)5.
1 2 ′
k − k ·
10