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Stability of the Blaschke-Santalo´ and the affine 9 0 isoperimetric inequality 0 2 n a Ka´roly J. Bo¨ro¨czky ∗ J 1 January 21, 2009 2 ] G M Dedicated toEndreMakaion theoccasionofhissixtiethbirthday . h t a Abstract m A stability version of the Blaschke-Santalo´ inequality and the affine [ 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. 1 0 9 2000MathematicsSubjectClassification: 52A40 0 : v i 1 Introduction X 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 classical 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 transformations 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´rańy [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 characterized the equality case among convex bodies with C2 boundary, and this characterization 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 transformations. 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 bodies 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 hyperplane. 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 statement. 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