JOHN S DECOMPOSITION OF THE IDENTITY IN THE NON-CONVEX CASE Jesus Bastero* ad Miguel Romace** Abstract We prove a extesio of the classical Joh s Theorem, that characterices the ellipsoid of maximal volume positio iside a covex body by the existece of some kid of decompositio of the idetity, obtaiig some results for maximal volume positio of a compact ad coected set iside a covex set with oempty iterior. By usig those results we give some estimates for the outer volume ratio of bodies ot ecesarily covex. 1. Itroductio ad Notatio Throughout this paper, we cosider R with the caoical basis (e 1,..., e ) ad its usual Euclidea structure,. Let B 2 = {x R ; x = x, x 1/2 1} be the euclidea ball o R. If K R, the it K, K c ad K will deote the iterior, the complemetary ad the border of K, respectively; cov(k) will be the covex hull of K, K 0 will deote the polar of K with respect to the origi, i.e. K 0 = {y R ; x, y 1, x K} ad vol (K) represets the Lebesgue measure o R of K. Followig [TJ], if K 1 K 2 R, we call a pair (x, y) R R a cotact pair for (K 1, K 2 ) if it satisfies: i) x K 1 K 2, ii) y K 0 2 ad iii) x, y = 1. As it is usual y x deotes the liear trasformatio o R defied by y x(z) = z, y x ad I will be the idetity map o R. Joh s ellipsoid theorem is a classical tool i the theory of covex bodies; it says how far a covex body is from beig a ellipsoid. Joh showed that each covex body cotais a uique ellipsoid of maximal volume ad characterized it. The decompositio of the idetity associated to this characterizatio gives a effective method to itroduce a appropiated euclidea structure i fiite dimesioal ormed spaces, whe we cosider cetrally symmetric covex bodies. We ca state Joh s theorem i the followig way: Theorem ([J], [Ba2], [Ba3]). Let K be a covex body i R ad suppose that the euclidea ball B2 is cotaied i K, the the followig assertios are equivalet: (i) B2 is the ellipsoid of maximal volume cotaied i K, (ii) there exist λ 1,..., λ m > 0 ad u 1,..., u m K B2, with m ( + 3)/2 such m m that I = u i u i ad λ i u i = 0. (iii) B 2 is the uique ellipsoid of maximal volume cotaied i K. * Partially supported by Grat D.G.E.S.I.C. (Spai) ** Supported by F.P.I. Grat D.G.E.S.I.C. (Spai) 1
Oe ca cosider this situatio for ay geeral couple of covex bodies (K 1, K 2 ) or, eve more, for ay couple of compact sets i R istead of (B 2, K). Suppose that K 1 is a compact set i R with vol (K 1 ) > 0 ad K 2 is aother compact set i R, with it K 2. A compactess argumet shows that there exists a affie positio of K 1, amely K 1, such that K 1 K 2 ad vol ( K 1 ) = max{vol (a + T (K 1 )); a + T (K 1 ) K 2, a R, T GL()}. This positio K 1 is called maximal volume positio of K 1 iside K 2. Very recetly, Giaopoulos, Perisiaki ad Tsolomitis have cosidered the covex situatio ad proved the followig Theorem [G-P-T]. Let K 1 K 2 be two smooth eough covex bodies i R such that K 1 is i maximal volume positio iside K 2. The for every poit z i the iterior of K 1, there exist λ 1,..., λ N > 0, with N 2 + + 1 ad cotact pairs for (K 1 z, K 2 z), (x i z, y i ), (1 i N) such that: (i) λ i y i = 0 ad (ii) I = λ i y i x i. Furthermore, if we assume the extra assumptio for K 1 to be a polytope ad K 2 to have C (2 boudary with strictly positive curvature, the a ceter z ca be chose i K 1 \ {vertices of K 1 } for which we have (i), (ii) ad also (iii) 1 λ i x i = z. The special case of cosiderig K 1 ad K 2 cetrally symmetric covex bodies was first observed by Milma (see [TJ], Theorem 14.5). The aim of this paper is to exted this result to the o-covex case. We obtai a geeral result which is valid for K 1 a compact, coected set i R with vol (K 1 ) > 0, K 1 K 2, where K 2 is a compact i R such that it cov(k 2 ) = ad K 1 is i maximal volume positio iside cov(k 2 ) (o extra assumptios o the boudary of the bodies are used). The method we develop to prove our result is differet from that i [G-P-T]. We follow the ideas give i [Ba3], with suitable modificatios ad the mai result we achieve is the followig Theorem 1.1. Let K 1 R be a coected, compact set with vol (K 1 ) > 0 ad K 2 R be a compact set such that K 1 K 2. If K 1 is i maximal volume positio iside cov(k 2 ), for every z it cov(k 2 ) there exist N N, N 2 +, (x i, y i ) cotact pairs for (K 1 z, K 2 z) ad λ i > 0 for all i = 1,..., N such that: λ k y k x k = 1 I λ k y k = 0. It is well kow that if there exists a decompositio of the idetity i the sese of theorem 1.1 we ca ot expect that K 1 were the uique maximal volume positio iside 2
K 2 eve for covex bodies, as it ca be show by cosiderig simpleces or octahedra iscribed i the cube. Furthermore, a equivalece as it appears i Joh s Theorem is ot true i geeral. We study this problem ad as a cosequece we obtai Theorem 1.2. Let K 1 R be a coected, compact set with vol (K 1 ) > 0 ad K 2 R be a compact set such that K 1 K 2. Let z be a fixed poit i it cov (K 2 ). The the followig asumptios are equivalets: (i) vol (K 1 ) = max{vol (a + S(K 1 )); a R, a + S(K 1 ) cov(k 2 )}, where S rus over all symmetric positive defiite matrices. (ii) There exist N N, N 2 +3 2, (x i, y i ) cotact pairs for (K 1 z, K 2 z) ad λ i > 0 for all i = 1,..., N such that: λ k (y k x k + x k y k ) = 1 I (1) λ k y k = 0. (iii) K 1 is the uique positio of K 1 verifyig (i). I sectio 3 we exted the upper estimates of the volume ratio proved i [G-P-T] by defiig the outer volume ratio of a compact K 1 with respect to a covex body K 2, by cosiderig a appropriate idex. We follow the methods that appears i [G-P-T] by usig Brascamp-Lieb ad reverse Brascamp-Lieb iequalities as the mai tools. 2. Proofs of Mai Theorems Throughout this sectio K 1 will be a coected, compact set i R with vol (K 1 ) > 0 ad K 2 will be a compact i R such that K 1 K 2. Therorem 1.1 gives us a sufficiet coditio for the existece of some kid of Joh s decompositio of the idetity ad it follows the spirit of the work of K.M. Ball (see for istace [Ba2] or [Ba3]). Proof of Theorem 1.1: First of all, otice that it cov (K 2 ), sice K 1 cov (K 2 ) ad vol (K 1 ) > 0. Without loss of geerality we ca assume that z = 0. Furthermore, sice K 1 K 2 ad cov(k 2 ) 0 = K2 0 a cotact pair for (K 1, cov(k 2 )) is also a cotact pair for (K 1, K 2 ), so, we may suppose cov(k 2 ) = K 2. Let A = {(y x, y) L(R, R ) R ; (x, y) is a cotact pair for (K 1, K 2 )}. By usig the maximality of the volume of K 1, the covexity of K 2 ad sice 0 it K 2 it is easy to prove that A is a o empty subset of L(R, R ) R. We will show that ( 1 I, 0) cov(a), where cov(a) is the covex hull of A i L(R, R ) R. Suppose that ( 1 I, 0) / cov(a). The by usig a separatio theorem, there exist H L(R, R ) ad b R such that: 1 I, H tr + 0, b > y x, H tr + b, y for all (x, y) cotact pair ad where, tr deotes the trace duality o L(R, R ), i.e. T, S tr = tr ST. 3
Thus for every cotact pair (x, y) 1 tr H > Hx, y + b, y. (2) There is o loss of geerality to assume tr H = 0. Ideed, we ca choose H = H I L(R, R ), which is a liear operator with trace zero that verifies: tr H Hx, y + b, y = Hx, y tr H x, y + b, y < 0 = tr H for all (x, y) cotact pair. By usig the liear map defied by the matrix H ad b R we are goig to costruct a family of affie maps S δ s, with 0 < δ < δ 1, such that det S δ 1 ad S δ (K 1 ) it K 2, which cotradicts the maximality of the volume of K 1. We will divide the proof of that fact ito 3 steps. By cotiuity, there exists a positive umber δ 0 > 0 such that I δh is ivertible for all 0 < δ < δ 0. For each 0 < δ < δ 0 we take S δ : R R defied by S δ (z) = (I δh) 1 (z) + δ(i δh) 1 (b). Step 1: There exists 0 < δ 1 δ 0 such that S δ (K 1 ) K 2 =, for all 0 < δ < δ 1. Cosider M = { x K 2 ; y K 0 2 such that x, y = 1 ad Hx, y + b, y 0 }. It is easy to check that M is a compact subset of K 2 ad also M K 1 =. If there exists a x M K 1 K 2 K 1, there would exist y K 0 2 such that x, y = 1 (so (x, y) is a cotact pair) ad Hx, y + b, y 0 = tr H which would cotradict (2). Therefore M K c 1; by compactess of M ad by cotiuity, there exists 0 < δ 1 ( δ 0 ) such that (I δh)(m) δb K c 1, for all 0 < δ < δ 1, ad so S δ (K 1 ) M =. Now let x K 2. We will prove that x / S δ (K 1 ). We oly have to cosider the case x / M. The Hx, y + b, y < 0 for all y K 0 2 such that x, y = 1. Sice 0 it K 2 ad K 2 is a covex body there exists y 0 K 0 2 such that x, y 0 = 1, so we have that x δhx δb, y 0 = 1 δ ( Hx, y 0 + b, y 0 ) > 1 for all δ > 0 ad i particular for all 0 < δ < δ 1. equivaletly x / S δ (K 1 ). Hece x δhx δb / K 1, or Step 2: For every 0 < δ < δ 1 there exists λ δ > 1 such that λ δ S δ (K 1 ) it K 2. Note that S δ (K 1 ) is coected ad S δ (K 1 ) K 2 =, therefore either S δ (K 1 ) it K 2 or S δ (K 1 ) K c 2. Fix x K 1 it K 2, it exists sice vol (K 1 ) = 0, ad take C x = {S δ (x) ; 0 δ < δ 1 }. It is easy to check that C x is coected, C x K 2 =, C x it K 2 ad C x it K 2. Therefore S δ (K 1 ) it K 2 ad by coectedess of K 1 we coclude that 4
S δ (K 1 ) it K 2, for all 0 < δ < δ 1. Now, by a compactess argumet ad the fact that S δ (K 1 ) it K 2 we coclude that for every 0 < δ < δ 1 there exists λ δ > 1 such that λ δ S δ (K 1 ) it K 2. Step 3: vol (λ δ S δ (K 1 )) > vol (K 1 ), for all 0 < δ < δ 1. Ideed, vol (λ δ S δ (K 1 )) = λ δ vol (K 1) det(i δh). Now, by usig the iequality betwee arithmetic mea ad geometric mea (deoted briefly AM-GM iequality) we obtai that det(i δh) 1 1 tr (I δh) = 1, ad so vol (λ δ S δ (K 1 )) λδ vol (K 1 ) > vol (K 1 ). Therefore, if ( 1 I, 0) / cova, the K 1 is ot i maximal volume positio iside K 2. Note that the fact that N 2 + is deduced from the classical Caratheodory s theorem sice {(y x 1 I, y) ; (x, y) is a cotact pair} is cotaied i a ( 2 + 1) dimesioal vector space. Remarks. 1) For every K R with vol (K) > 0 it is easy to check that K is i maximal volume positio iside cov (K). 2) We ote that K 1 K 2 ad K 1 is i maximal volume positio iside cov (K 2 ) implies that K 1 is i maximal volume positio iside K 2. The coverse is ot true, as the followig example shows. Cosider K 1 = {x R ; x = max 1 i x i 1} K 2 = K 1 2 K 1. It is trivial to see that K 1 is i maximal volume positio iside K 2, K 1 is ot i maximal volume positio iside cov(k 2 ) ad there is o decompositio of the idetity as before, sice there are o cotact pairs for (K 1, K 2 ). Corolary 2.1. Let K 1 K 2 R be as i the theorem 1.1. If cov (K 1 ) is a polytope, cov (K 2 ) has C (2) boudary with strictly positive curvature ad K 1 is maximal volume positio iside cov (K 2 ), the there exist z cov (K 1 ), N N, N 2 +, (x k, y k ) cotact pairs of (K 1 z, K 2 z) ad λ k > 0 for all k = 1,..., N such that λ k y k x k = 1 I λ k y k = λ k x k = 0. Proof: It is easy to prove that the fact that K 1 is i maximal volume positio iside cov (K 2 ) implies that cov (K 1 ) is i maximal volume positio iside cov (K 2 ) ad K 1 (cov (K 2 )) = (cov (K 1 )) (cov (K 2 )). Therefore we ca assume that 5
K 1 is a polytope ad K 2 is a covex which has a C (2) boudary with strictly positive curvature. But otice that this case was studied by A. Giaopoulos, I. Perissiaki ad A. Tsolomitis (see [G-P-T]) cocludig the result eeded. Now we ca ask if the existece of some kid of decompositio of the idetity i R would imply that K 1 were the uique maximal volume positio iside K 2, as it happes i the classical Joh s Theorem. It is well kow that we ca t expect such a thig, simply by cosiderig simplices or octahedra iscribed i the cube. Theorem 1.2 shows, loosely speakig, that ot oly the existece of a modified Joh s decompositio of the idetity for a pair (K 1, K 2 ) implies that K 1 is the uique pseudo maximal volume positio iside K 2, but also that this pseudo maximality implies the existece of a modified decompositio of the idetity too. Proof of Theorem 1.2: As before, we ca show that it cov (K 2 ). We ca also assume z = 0 ad K 2 covex. (i) (ii) Let B = {( 1 2 (y x + x y), y) L(R, R ) R ; (x, y) is a cotact pair}. By usig the maximality of the volume of K 1, the covexity of K 2 ad sice 0 it K 2 it is easy to prove that B is a o empty subset of L(R, R ) R. As i the proof of theorem 1.1, we will show that ( 1 I, 0) cov (B). Suppose, o the cotrary, that ( 1 I, 0) / cov (B). The by usig a separatio theorem, there exist H L(R, R ) ad θ R such that: 1 I, H tr + 0, θ > 1 2 ( y x, H tr + x y, H tr ) + θ, y for all (x, y) cotact pair. Therefore 1 tr H > 1 ( Hx, y + x, Hy ) + θ, y. 2 There is o loss of geerality to assume that: 1) H is a symmetric matrix because i other case we could take H = 1 2 (H + H ) which is a symmetric matrix that verifies that Hx, y + x, Hy = Hx, y + x, Hy ad therefore for every cotact pair (x, y) 1 tr H > Hx, y + θ, y. 2) tr H = 0 sice i other case we ca choose H = H tr H I L(R, R ) which is a liear operator with trace zero that verifies: Hx, y + θ, y = Hx, y tr H x, y + θ, y < 0 = tr H for all (x, y) cotact pair. Therefore there exist a symmetric matrix H L(R, R ) with tr H = 0 ad θ R such that: 0 > Hx, y + θ, y 6
for all (x, y) cotact pair. By usig the liear map defied by the matrix H ad θ R we are goig to costruct a family of affie maps T δ ( ) = S δ ( ) + b δ, with S δ symmetric positive defiite matrix for all 0 < δ < δ 1, such that det S δ 1 ad T δ (K 1 ) it K 2, which cotradicts the maximality of K 1. By cotiuity, there exists a positive umber δ 0 > 0 such that I δh is ivertible ad symmetric positive defiite for all 0 < δ < δ 0. For each 0 < δ < δ 0 we take T δ : R R defied by T δ (z) = (I δh) 1 (z) + δ(i δh) 1 (θ). By the same methods as i the proof of Theorem 1.1, we ca show that: 1) There exists 0 < δ 1 δ 0 such that T δ (K 1 ) K 2 =, for all 0 < δ < δ 1. 2) For every 0 < δ < δ 1 there exists λ δ > 1 such that λ δ T δ (K 1 ) it K 2. 3) vol (λ δ T δ (K 1 )) > vol (K 1 ), for all 0 < δ < δ 1 which cotradicts the maximality of K 1. (ii) (iii) Let T ( ) = S( ) + a be such that T (K 1 ) K 2, a R ad S is a symmetric positive defiite matrix. It is well kow that we ca fid a orthogoal matrix U O() ad a diagoal matrix D with diagoal elemets α 1,..., α > 0 such that S = U D U ad therefore ( ) vol (T (K 1 )) = det (U D U) vol (K 1 ) = α k vol (K 1 ). (3) Hece we have to estimate α k. O the oe had, we obtai that U D Ux, y = α j U e j, x U e j, y j=1 for all x, y R, by straightforward computatio. O the other had, if (x, y) is a cotact pair the T x, y 1 ad therefore 1 = = = 1 λ k λ k λ k T x k, y k = j=1 λ k U D Ux k, y k α j U e j, x k U e j, y k = α j U e j, U e j = 1 j=1 α j. j=1 ( j=1 α j N λ k U e j, x k U e j, y k Now by usig the AM-GM iequality, we coclude that 1 1 αj ( α j ) 1, which implies that i (3) we obtai vol (T (K 1 )) vol (K 1 ). I additio to this, ote that if T is such that vol (T (K 1 )) = vol (K 1 ), the, by the equality case i the AM-GM iequality we would have that α 1 =... = α = 1, so T = I + a. Therefore we would obtai that 1 T x, y = x + a, y = 1 + a, y 7 )
for all (x, y) cotact pair ad, i particular, a, y k 0 for all (x k, y k ) cotact pair that appears i the decompositio of the idetity. But we also would have that: λ k a, y k = a, λ k y k = 0 which would imply that, a, y k = 0 for all (x k, y k ) ad the we would coclude that Hece T =I. 1 N a, a = λ k a, y k a, x k = 0. Corolary 2.2. Let K 1 K 2 be as i theorem 1.1. Fix z it cov (K 2 ). The the followig assumptios are equivalets: (i) vol (K 1 ) = max{vol (a + S(K 1 )); a R, a + S(K 1 ) cov(k 2 )}, where S rus over all symmetric positive defiite matrices. (ii) For every S GL() symmetric matrix ad every θ R there exists a cotact pair (x, y) for (K 1 z, K 2 z) such that tr S Sx, y + θ, y. Proof: As before, we ca show that it cov (K 2 ). We ca also assume z = 0 ad K 2 covex. (i) (ii) By Theorem 1.2 there exist (x i, y i ) cotact pairs for (K 1 z, K 2 z) ad λ i > 0 for all i = 1,..., N such that: λ k (y k x k + x k y k ) = 1 I ad λ k y k = 0. Suppose that there would exist S GL() symmetric matrix ad θ R such that for every (x, y) cotact pair Therefore tr S > Sx, y + θ, y. tr S = tr S = < N + θ, λ i y i = (S, θ), ( 1 I, λ k y k ) = λ i (S, θ), ( 1 2 (y i x i + x i y i ), y i ) = λ i tr S = tr S λ i ( Sx i, y i + θ, y k ) < 8
which leads us to a cotradictio. (ii) (i) By usig the hypothesis, for every H GL(), θ R, there exists a cotact pair such that 1 tr H > 1 ( Hx, y + x, Hy ) + θ, y 2 which make that ( 1, 0) cov ({( 1 2 (y x + x y), y) L(R, R ) R ; where (x, y) is a cotact pair }) Remarks. 1) If K 1 is i maximal volume positio iside cov (K 2 ), the K 1 is uique if we oly cosider affie trasformatios give by symmetric, positive defiite matrices. Ideed, this is due to the fact that 1 I = N λ k y k x k implies that 1 I = N λ k x k y k. 2) If we suppose either K 1 =B2 or cov(k 2 )=B2 i the last theorem, we obtai a stroger coclusio, sice the existece of cotact pairs (x k, y k ) ad λ k > 0 such that λ k 2 (y k x k + x k y k ) = 1 I ad λ k y k = 0 is equivalet to the fact that vol (K 1 ) = max{vol (a + T (K 1 )) such that a + T (K 1 ) cov (K 2 ), a R ad T GL()} ad this maximum is oly attaied at K 1, up to orthogoal trasformatio (i.e. if vol (T (K 1 )) = vol (K 1 ), the T is a orthogoal trasformatio). This is the classical Joh s result. Let s see it briefly. Suppose that there exists a decompositio of the idetity (i the sese of (1)). If we take cov(k 2 ) = B 2 ad T is a affie trasformatio such that T (K 1 ) B 2, the there exist orthogoal matrices U, V, a diagoal matrix D with diagoal elemets α 1,..., α > 0 ad a R such that T ( ) = V DU( ) + a. Now if we choose T ( ) = U DU( ) + (V U) (a) the it is easy to check that this map verifies: (a) U DU is a symmetric positive defiite matrix. (b) T (K 1 ) (V U) (B 2 ) = B 2 (sice T ( ) = (V U) T ( )). (c) vol ( T (K 1 )) = vol (T (K 1 )). Therefore by usig (ii) (iii) i theorem 1.2 ad sice T satisfies (a) ad (b) we coclude that vol (T (K 1 )) = vol ( T (K 1 )) vol (K 1 ) ad the equality is oly attaied if T = I, ad so T is a orthogoal trasformatio. Note that a similar reasoig ca be applied to the case K 1 = B 2. 9
3. Some estimates for the outer volume ratio of compact sets We ca exted the otio of volume ratio to a pair (K 1, K 2 ) R R, where K 2 is a covex body ad K 1 is a compact set with vol (K 1 ) > 0, simply by Defiitio 3.1. Let K 1 R be compact set with vol (K 1 ) > 0 ad K 2 R be a covex body. We defie outer volume ratio as { vol (K 2 ) 1 vr(k 2 ; K 1 ) = if ; T affie trasformatio with T (K vol (T (K 1 )) 1 1 ) K 2 }. It is quite easy to show that we caot expect ay upper estimate without asumig extra asumptios. We are goig to itroduce a idex for compact sets with positive volume i order to get geeral bouds, depedig oly o the dimesio ad o the idex, for the outer volume ratio with respect to a covex body. We recall that a set K R is p-covex, (0 < p 1) if λx + µy K, for every x, y K ad for every λ, µ 0 such that λ p + µ p = 1. The p-covex hull of a set K, which we deote by p cov (K), is defied as the itersectio of all p-covex sets that cotai K. It is easy to see that 0 p cov (K). Defiitio 3.2. Let K R a compact set. We defie p(k) as p(k) = { sup {p (0, 1] ; a R with p cov{(extk) a} K a} if it exists 0 otherwise where extk deotes the set of extreme poits of K. Remarks: 1) If p (0, 1] verifies that there exist a a R such that p cov{(extk) a} K a the a K, sice 0 is iside the clausure of p cov{(extk) a}, which is embedded i K a ad so a K. 2) p(k) is a affie ivariat of K, i.e. if T = a + S is a affie trasformatio o R with a R ad S GL() the p(t (K)) = p(k). 3) The supremum i the last defiitio ca be replaced by maximum, simply by usig compactess ad cotiuity argumets. 4)If K is a p-covex body with 0 < p 1 the p(k) p, but if 0 < p < 1 the there are compact sets K with p(k) p which are ot p-covex. Notice that p(k) = 1 if ad oly if K is covex, simply by usig Krei-Milma s theorem. Now we are goig to state ad prove some upper estimates for the volume ratio of a pair (K 1, K 2 ) where K 1 is a compact set with vol (K 1 ) > 0 ad p(k 1 ) > 0, ad K 2 is a covex body. We ca assume that K 1 is i maximal volume positio iside K 2, sice i other case, there would exist a affie trasformatio T such that T (K 1 ) would be i maximal volume positio iside K 2 ad therefore K 1 would work with the pair (T (K 1 ), K 2 ). Hece if p(k 1 ) = p, vr(k 2 ; K 1 ) = vol (K 2) 1 vol (K 1 ) 1 vol (K 2 ) 1 vol (p cov {(extk 1 ) a}) 1 10
for some a K 1. Therefore vr(k 2 ; K 1 ) vol (K 2 a) 1 vol (cov {(extk 1 ) a}) 1 vol (cov {(extk 1 ) a}) 1. vol (p cov {(extk 1 ) a}) 1 It ca be show that cov {(extk 1 ) a} = cov (K 1 a) ad sice cov (K 1 a) is i maximal volume positio iside K 2 a we get vol (cov {(extk 1 ) a}) 1 vr(k 2 ; K 1 ) vr(k 2 ; cov (K 1 )). vol (p cov {(extk 1 ) a}) 1 It is easy to check that cov {(extk 1 ) a} 1 p 1 (p cov {(extk) a}). Ideed, sice a K 1 the cov {(extk 1 ) a} = cov (K 1 a) = cov { x K1 [0, x a]} ad we ca use a stroger versio of Caratheodory s theorem appearig i [E] that asserts that for every x cov {(extk 1 ) a} there exist x i (extk) a ad α i 0, i = 1,..., such that x = α ix i ad α i = 1. Therefore ( α p i ) 1 p 1 p 1 which implies that x 1 p 1 p cov {(extk 1 ) a}. O the other had a result of Giaopoulos, Perissiaki ad Tsolomitis (see [G-P-T]) shows that vr(k 2 ; cov {(extk 1 ) a}) ad thus we sumarize all these thigs i the followig result Propositio 3.3. Let K 1, K 2 R be such that K 1 is a compact set with vol (K 1 ) > 0, p(k 1 ) = p > 0 ad K 2 a covex body. The vr(k 2 ; K 1 ) 1 p. Next we are goig to prove that if K 1 or K 2 has some kid of symmetry properties the this geeral estimate ca be slightly improved by usig decompositios of the idetity i the sese of theorem 1.1, followig the spirit of K.M. Ball (see [Ba1]) ad A. Giaopoulos, I. Perisiaki, A. Tsolomitis ([G-P-T]). We start with a result which ca be foud i [G-P-T] ad whose proof ivolves Cauchy-Biet formula. Lemma 3.4. Let λ 1,..., λ N > 0. Let x 1,..., x N ad y 1,..., y N be vectors i R satisfyig x k, y k = 1, for all k = 1,..., N ad λ k y k x k = I. The D x D y 1, where D x ad D y are defied by { det( } N D x = if λ kα k x k x k ) ; α k > 0, k = 1,..., N N αλ k k 11 α i, (4)
{ det( } N D y = if λ kα k y k y k ) ; α k > 0, k = 1,..., N. (5) N αλ k k Propositio 3.5. Let K 1, K 2 R be such that K 1 is a symmetric compact set with vol (K 1 ) > 0, p(k 1 ) = p > 0 ad K 2 is a symmetric covex body. The vr(k 2 ; K 1 )! 1 1 p 1. Proof: First of all it is easy to check that we ca assume that K 1 ad K 2 are cetrally symmetric ad so it is ext K 1. By usig the same argumets tha before we coclude that vr(k 2 ; K 1 ) vr(k 2 ; cov (K 1 )) 1/p 1. Next we are goig to give a upper estimate for vr(k 2 ; L), where K 2 ad L = cov (K 1 ) are cetrally symmetric covex bodies ad L is i maximal volume positio iside K 2. By usig theorem 1.1, we ca fid cotact pairs (x i, y i ) ad λ i > 0, for all i = 1,..., N, N 2 +, such that λ k y k x k = I ad λ k y k = 0. If we take X = cov {±x 1,..., ±x N } L ad Y = {y R ; y, y k 1 k = 1,..., N} K 2 Y, we obtai that vr(k 2 ; L) = vol (K 2) 1 vol (L) 1 vol (Y ) 1. vol (X) 1 Therefore if we fid some upper estimate for vol (Y ) ad lower estimate for vol (X) we will obtai some upper estimates for vr(k 2 ; K 1 ). Claim 1: vol (Y ) 2 Dy Cosider g j : R R, j = 1,..., N, defied by g j (t) = χ [ 1,1] (t). By usig the Brascamp-Liev iequality (see [Bar]) we obtai that R N (g k ( x, y k )) λ k dx 1 Dy N ( R ) λk g k (t) dt = 1 ( 1 Dy 1 ) λ k dt where D y was defied i (5). O the other had, we coclude that Therefore vol (Y ) R N (g k ( x, y k )) λ k dx = χ Y (x) dx = vol (Y ). R 2 Dy 12
Claim 2: vol (X) 2 D x! We defie for every x R { N N(x) = if α k ; x = } α k x k which is a itegrable fuctio that verifies { N } e N(x) dx = sup e αp k ; αk 0, x = α k x k dx R R { N } = sup f k (θ k ) λ k ; x = λ k θ k x k dx R where f k : R R is defied by f k (t) = e t. Now, if we use the reverse of the Brascamp-Liev iequality (see [Bar]) we ca assert that { N sup f k (θ k ) λ k ; x = R } λ k θ k x k dx D x N = D x N = D x 2 ( 2 λ k R ) λk f k (t) dt where D x was defied i (4). O the other had, we ca compute directly the itegral of e N(x) by e N(x) dx = R R + N(x) e t dt dx = + 0 e t {N(x) t} dxdt. It is easy to check that {x R ; N(x) t} = tx, for all t > 0, ad hece R e N(x) dx = + 0 (6) e t t vol (X) dt =! vol (X). (7) So, combiig (6) ad (7) we coclude the desired lower estimate for vol (X) ad by usig Claim 1, Claim 2 ad lemma 3.4 we obtai that ad hece, the result holds. vr(k 2 ; L)! 1 By usig similar argumets we ca prove the followig result 13
Propositio 3.6. Let K 1, K 2 R are such that K 1 is a compact set with vol (K 1 ) > 0, p(k 1 ) = p > 0 ad K 2 is a covex body, the: (1) If K 1 is symmetric, vr(k 2 ; K 1 ) vr(k 2 ; K 1 ) e 2 (!) 1 1 p 1. (2) If K 2 is symmetric,vr(k 2 ; K 1 ) vr(k 2 ; K 1 ) 2 (!) 1 1 p 1. Proof: (1) Take g j (t) = e t χ (,1] (t) istead of g j (t) i the proof of propositio 3.5. (2) Take f j (t) = e t χ [0,+ ) (t) istead of f j (t) i the proof of propositio 3.5 ad substitute N(x) by Ñ(x) = { N } if α k ; α k 0, x = α k x k if it exists + otherwise. Refereces [Ba1] K.M. Ball, Volume Ratios ad a reverse isoperimetric iequality, J. Lodo Math. Soc. (2) 44 (1991), pp. 351-359. [Ba2] K.M. Ball, Ellipsoids of Maximal Volume i covex bodies, Geometria Dedicata 41 (1992), pp. 241-250. [Ba3] K.M. Ball, A Elemetary Itroductio to Moder Covex Geometry i Flavours of Geometry. Edited by S. Levy. Cambridge Uiversity Press. (1997), pp. 1-55. [Bar] F. Barthe, Iégalités de Brascamp-Liev et covexité, C.R. Acad. Sci. Paris 324 (1997), pp. 885-888. [E] H.G. Egglesto, Covexity, Cambridge Tracts i Math., vol. Uiv. Press, Lodo ad New York (1969). 47, Cambridge [G-P-T] A. Giaopoulos, I. Perissiaki ad A. Tsolomitis, Joh s Theorem for a arbitrary pair of covex bodies, preprit. [J] F. Joh, Extremum problems with iequalities as subsidiary coditios, Courat Aiversary Volume, New York (1948), pp. 187-204 [TJ] N. Tomczak-Jaegerma, Baach-Mazur distaces ad fiite dimesioal operator ideals, Pitma Moographs 38 (1989), Pitma, Lodo. Jesus Bastero: departameto de matemáticas, uiversidad de zaragoza, 50009-zaragoza, spai. e-mail:bastero@posta.uizar.es Miguel Romace: departameto de matemáticas, uiversidad de zaragoza, 50009-zaragoza, spai. e-mail:mromace@posta.uizar.es 14