Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries
aa r X i v : . [ m a t h . M G ] J a n Stability of the log-Brunn-Minkowski inequality inthe case of many hyperplane symmetries
K´aroly J. B¨or¨oczky ∗† , Apratim De ‡ January 12, 2021
Dedicated to Prof. Erwin Lutwak on the occasion of his seventy-fifthbirthday
Abstract
In the case of symmetries with respect to n independent linearhyperplanes, a stability version of the logarithmic Brunn-Minkowskiinequality and the logarithmic Minkowski inequality for convex bodiesis established. MSC
The classical Brunn-Minkowski inequality form the core of various areasin probability, additive combinatorics and convex geometry (see Gardner[38], Gruber [44], Schneider [69] and Tao, Vu [70]). For recent related workin the theory of valuations, algorithmic theory and the Gaussian setting,see say Jochemko, Sanyal [48, 49], Kane [50], Gardner, Gronchi [39] andGardner, Zvavitch [40]. Extending it, the rapidly developing new L p -Brunn-Minkowski theory initiated by Lutwak [55, 56, 57], have become main re-search area in modern convex geometry and geometric analysis. ∗ Supported by NKFIH grant K 132002 † Alfr´ed R´enyi Institute of Mathematics, Realtanoda u. 13-15, H-1053 Budapest, Hun-gary, and Department of Mathematics, Central European University, Nador u. 9, H-1051,Budapest, Hungary, [email protected] ‡ Department of Mathematics, Central European University, Nador u. 9, H-1051, Bu-dapest, Hungary, [email protected] L p -Brunn-Minkowski inequality for arange of p by Firey [36] and Lutwak [55, 56, 57], major results have beenobtained by Hug, Lutwak, Yang, Zhang [46], and more recently the pa-pers Kolesnikov, Milman [52], Chen, Huang, Li, Liu [20], Hosle, Kolesnikov,Livshyts [45], Kolesnikov, Livshyts [51] present new developments and ap-proaches.We call a compact compact set K in R n a convex body if V ( K ) > V ( K ) stands for the n -dimensional Lebesgue measure. The corner-stone of the Brunn-Minkowski Theory is the Brunn-Minkowski inequality(see Schneider [69]). If K and C are convex bodies in R n and α, β >
0, thenthe Brunn-Minkowski inequality says that V ( αK + βC ) n ≥ αV ( K ) n + βV ( C ) n (1)where equality holds if and only if C = γK + z for γ > z ∈ R n .Because of the homogeneity of the Lebesgue measure, (1) is equivalent tosay that if λ ∈ (0 , V ((1 − λ ) K + λ βC ) n ≥ V ( K ) − λ V ( C ) λ (2)where equality holds if and only if K and C are translates. We also noteanother consequence of the Brunn-Minkowski inequality (1); namely, theMinkowski inequality says that Z S n − h C dS K ≥ Z S n − h K dS K provided V ( C ) = V ( K ) . (3)We note that the L p -Minkowski and L p -Brunn-Minkowski inequalities areextended to certain families of non-convex sets by Zhang [74], Ludwig, Xiao,Zhang [54] and Lutwak, Yang, Zhang [58].The first stability forms of the Brunn-Minkowski inequality were dueto Minkowski himself (see Groemer [43]). If the distance of K and C ismeasured in terms of the so-called Hausdorff distance, then Diskant [25] andGroemer [42] provided close to be optimal stability versions (see Groemer[43]). However, the natural distance is in terms volume of the symmetricdifference, and the optimal result is due to Figalli, Maggi, Pratelli [32, 33].To define the “homothetic distance” A ( K, C ) of convex bodies K and C , let α = | K | − n and β = | C | − n , and let A ( K, C ) = min {| αK ∆( x + βC ) | : x ∈ R n } K ∆ Q stands for the symmetric difference of K and Q . In addition,let σ ( K, C ) = max n | C || K | , | K || C | o . Now Figalli, Maggi, Pratelli [33] proved thatsetting γ ∗ = ( (2 − n − n ) n ) , we have | K + C | n ≥ ( | K | n + | C | n ) " γ ∗ σ ( K, C ) n · A ( K, C ) . Here the exponent 2 of A ( K, C ) is optimal ( cf. Figalli, Maggi, Pratelli[33]). We note that prior to [33], the only known error term in the Brunn-Minkowski inequality was of order A ( K, C ) η with η ≥ n , due to Diskant[25] and Groemer [42] in their work on providing stability result in termsof the Hausdorff distance (see also Groemer [43]), and also to a more di-rect approach by Esposito, Fusco, Trombetti [29]; therefore, the exponentdepended significantly on n .We note that recently, various breakthrough stability results about ge-ometric functional inequalities have been obtained. Fusco, Maggi, Pratelli[37] proved an optimal stability version of the isoperimetric inequality (whoseresult was extended to the Brunn-Minkowski inequality by Figalli, Maggi,Pratelli [32, 33], see also Eldan, Klartag [28]). Stonger versions of the Borell-Brascamp-Lieb inequality are provided by Ghilli, Salani [41] and Rossi,Salani [66], and of the Sobolev inequality by Figalli, Zhang [35] (extend-ing Bianchi, Egnell [10] and Figalli, Neumayer [34]), Nguyen [61] and Wang[73], and of some related inequalities by Caglar, Werner [17]. Related in-equalities are verified by Colesanti [21], Colesanti, Livshyts, Marsiglietti [22],P. Nayar, T. Tkocz [59, 60], Xi, Leng [71].In this paper, we focus on the L sum of replacing Minkowski addition.First, for λ ∈ (0 , L or logarithmic sum of two origin symmetricconvex bodies K and C in R n is defined by(1 − λ ) · K + λ · C = n x ∈ R n : h x, u i ≤ h K ( u ) − λ h C ( u ) λ ∀ u ∈ S n − o . It is linearly invariant, as A ((1 − λ ) · K + λ · C ) = (1 − λ ) · A K + λ · A C for A ∈ GL( n ). The following strengthening of the Brunn-Minkowski inequality forcentered convex bodies is a long-standing and highly investigated conjecture. CONJECTURE 1.1 (Logarithmic Brunn-Minkowski conjecture) If λ ∈ (0 , and K and C are convex bodies in R n whose centroid is the origin,then V ((1 − λ ) · K + λ · C ) ≥ V ( K ) − λ V ( C ) λ , (4)3 ith equality if and only if K = K + . . . + K m and C = C + . . . + C m compact convex sets K , . . . , K m , C , . . . , C m of dimension at least one where P mi =1 dim K i = n and K i and C i are dilates, i = 1 , . . . , m . We note that the choice of the right translates of K and C are importantin Conjecture 1.1 according to the examples by Nayar, Tkocz [59]. Onthe other hand, the following is an equivalent form of the origin symmetriccase of the Logarithmic Brunn-Minkowski conjecture for o -symmetric convexbodies. CONJECTURE 1.2 (Logarithmic Minkowski conjecture) If K and C are convex bodies in R n whose centroid is the origin, then Z S n − log h C h K dV K ≥ V ( K ) n log V ( C ) V ( K ) with the same equality conditions as in Conjecture 1.1. We note that understanding the equality case in Conjecture 1.2 clari-fies the uniqueness of the solution of the Monge-Ampere type logarithmicMinkowski Problem (see Boroczky, Lutwak, Yang, Zhang [15], Kolesnikov,Milman [52], Chen, Huang, Li, Liu [20]).In R , Conjecture 1.1 is verified in Boroczky, Lutwak, Yang, Zhang [15]for o -symmetric convex bodies, but it is still open in general. On the otherhand, Xi, Leng [71] proved that any two dimensional convex bodies K and C in R can be translated in a way such that (4) holds for the translates.In higher dimensions, Conjecture 1.1 is proved for with enough hyperplanesymmetries ( cf. Theorem 1.3) and complex bodies ( cf.
Rotem [67]).For o -symmetric convex bodies, Conjecture 1.2 is proved when K is closeto be an ellipsoid by a combination of the local estimates by Kolesnikov, Mil-man [52] and the use of the continuity method in PDE by Chen, Huang, Li,Liu [20]. Another even more recent proof of this result based on Alexan-drov’s approach of considering the Hilbert-Brunn-Minkowski operator forpolytopes is due to Putterman [65]. Additional local versions of Conjec-ture 1.2 for o -symmetric convex bodies are due to Kolesnikov, Livshyts [51].We say that A ∈ GL( n ) is a linear reflection associated to the linear( n − H ⊂ R n if A fixes the points of H and det A = −
1. In thiscase, there exists u ∈ R n \ H such that Au = − u where the invariant subspace R u is uniquely determined (see Davis [24], Humphreys [47], Vinberg [72]).It follows that a linear reflection A is a classical ”orthogonal” reflection ifand only if A ∈ O ( n ). 4ollowing the result on unconditional convex bodies by Saroglou [68],Boroczky, Kalantzopoulos [14] verified the logarithmic Brunn-Minkowskiand Minkowski conjectures under hyperplane symmetry assumption. THEOREM 1.3 (Boroczky, Kalantzopoulos)
If the convex bodies K and C in R n are invariant under linear reflections A , . . . , A n through n hyperplanes H , . . . , H n with H ∩ . . . ∩ H n = { o } , then V ((1 − λ ) · K + λ · C ) ≥ V ( K ) − λ V ( C ) λ (5) Z S n − log h C h K dV K ≥ V ( K ) n log V ( C ) V ( K ) , (6) with equality in either inequality if and only if K = K + . . . + K m and C = C + . . . + C m for compact convex sets K , . . . , K m , C , . . . , C m of dimensionat least one and invariant under A , . . . , A n where K i and C i are dilates, i = 1 , . . . , m , and P mi =1 dim K i = n . Geometric inequalities under n independent hyperplane symmetries werefirst considered by Barthe, Fradelizi [7] and Barthe, Cordero-Erausquin [6].These papers verified the classical Mahler conjecture and Slicing conjecture,respectively, for these type of bodies. The main result of our paper is astability version of Theorem 1.3. THEOREM 1.4 If λ ∈ [ τ, − τ ] for τ ∈ (0 , ] , the convex bodies K and C in R n are invariant under linear reflections A , . . . , A n through n hyper-planes H , . . . , H n with H ∩ . . . ∩ H n = { o } , and V ((1 − λ ) · K + λ · C ) ≤ (1 + ε ) V ( K ) − λ V ( C ) λ for ε > , then for some m ≥ , there exist compact convex sets K , C , . . . , K m , C m of dimension at least one and invariant under A , . . . , A n where K i and C i are dilates, i = 1 , . . . , m , and P mi =1 dim K i = n such that K + . . . + K m ⊂ K ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( K + . . . + K m ) C + . . . + C m ⊂ C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( C + . . . + C m ) where c > is an absolute constant. / (95 n ) shouldbe at least 1 /n . If for small ε > K is obtained from the box K =[ − n − , n − ] × [ − , n − by cutting off corners of size of order ε n , and C isobtained from the box C = [ − n − , n − ] × [ − , ] n − by cutting off cornersof suitable size of order ε n , then · K + · C = [ − , n , and V (cid:18) · K + · C (cid:19) ≤ (1 + ε ) V ( K ) V ( C ) , but if ηK ⊂ K for η >
0, then η ≤ − γ ε n where γ > n .We deduce from Theorem 1.4 a stability version of the logarithmic-Minkowski inequality (6) for convex bodies with many hyperplane symme-tries. THEOREM 1.5
If the convex bodies K and C in R n are invariant underlinear reflections A , . . . , A n through n hyperplanes H , . . . , H n with H ∩ . . . ∩ H n = { o } , and Z S n − log h C h K dV K V ( K ) ≤ n · log V ( C ) V ( K ) + ε for ε > , then for some m ≥ , there exist compact convex sets K , C , . . . , K m , C m of dimension at least one and invariant under A , . . . , A n where K i and C i are dilates, i = 1 , . . . , m , and P mi =1 dim K i = n such that K + . . . + K m ⊂ K ⊂ (cid:16) c n ε n (cid:17) ( K + . . . + K m ) C + . . . + C m ⊂ C ⊂ (cid:16) c n ε n (cid:17) ( C + . . . + C m ) where c > is an absolute constant. To prove Theorem 1.4, first we verify it in the unconditional case, seeSection 2 presenting these partial results. More precisely, first we considerthe coordinatewise product of unconditional convex bodies based on therecent stability version of the Prekopa-Leindler inequality (see Section 3),and then handle the unconditional case Theorem 2.3 of Theorem 1.4 inSections 4 and 5. Next we review some fundamental properties of Weylchambers and Coxeter groups in general in Section 6 and Section 7, andprove Theorem 1.4 in Section 8. Finally, Theorem 1.5 is verified in Section 9.6
The case of unconditional convex bodies
The way to prove Theorem 1.4 is first clarifying the case of unconditionalconvex bodies; namely, when A , . . . , A n are orthogonal reflections and H , . . . , H n are coordinate hyperplanes. For unconditional convex bodies, the coordi-natewise product is a classical tool; namely, if λ ∈ (0 ,
1) and K and C areunconditional convex bodies in R n , then K − λ · C λ = { ( ±| x | − λ | y | λ , . . . , ±| x n | − λ | y n | λ ) ∈ R n :( x , . . . , x n ) ∈ K and ( y , . . . , y n ) ∈ C } . It is known that (see say Saroglou [68]) that K − λ · C λ is a convex uncon-ditional body, and it follows from the H¨older inequality (see also Saroglou[68]) that K − λ · C λ ⊂ (1 − λ ) · K + λ · C. In addition, [68] verifies that if λ ∈ (0 , T is a positive definite diagonalmatrix and K is an unconditional convex body in R n , then K − λ · ( T K ) λ = T λ K (7)where T η = ( t η , . . . , t ηn ) for η ∈ R and T = ( t , . . . , t n ) for t , . . . , t n > V ((1 − λ ) · K + λ · C ) ≥ V ( K ) − λ V ( C ) λ in (8) about the coordinatewise product,even before the log-Brunn-Minkowski conjecture was stated, and the con-tainment relation between the coordinatewise product and the L -sum andthe description of the equality case are due to Saroglou [68]. For X, Y ⊂ R n ,we write X ⊕ Y to denote X + Y if lin X and lin Y are orthogonal. THEOREM 2.1 (Saroglou) If K and C are unconditional convex bodiesin R n and λ ∈ (0 , , then V ((1 − λ ) · K + λ · C ) ≥ V ( K − λ · C λ ) ≥ V ( K ) − λ V ( C ) λ . (8) (i) V ( K − λ · C λ ) = V ( K ) − λ V ( C ) λ if and only if C = Φ K for a positivedefinite diagonal matrix Φ . (ii) V ((1 − λ ) · K + λ · C ) = V ( K ) − λ V ( C ) λ if and only if K = K ⊕ . . . ⊕ K m and L = L ⊕ . . . ⊕ L m for unconditional compact convexsets K , . . . , K m , L , . . . , L m of dimension at least one where K i and L i are dilates, i = 1 , . . . , m .
7e note that the second inequality in (8) (about the coordinatewiseproduct) is a consequence of the Prekopa-Leindler inequality (see Section 3).In turn, the stability version Proposition 3.2 of the Prekopa-Leindler inequal-ity yields the following:
THEOREM 2.2 If λ ∈ [ τ, − τ ] for τ ∈ (0 , ] , and the unconditionalconvex bodies K and C in R n satisfy V ( K − λ · C λ ) ≤ (1 + ε ) V ( K ) − λ V ( C ) λ for ε > , then there exists positive definite diagonal matrix Φ such that V ( K ∆(Φ C )) < c n n n (cid:16) ετ (cid:17) V ( K ) and V ((Φ − K )∆ C ) < c n n n (cid:16) ετ (cid:17) V ( C ) where c > is an absolute constant. In the case of the logarithmic-Brunn-Minkowski inequality for uncondi-tional convex bodies, we have a different type stability estimate:
THEOREM 2.3 If λ ∈ [ τ, − τ ] for τ ∈ (0 , ] , and the unconditionalconvex bodies K and C in R n satisfy V ((1 − λ ) · K + λ · C ) ≤ (1 + ε ) V ( K ) − λ V ( C ) λ for ε > , then for some m ≥ , there exist θ , . . . , θ m > and uncondi-tional compact convex sets K , . . . , K m such that lin K i , i = 1 , . . . , m , arecomplementary coordinate subspaces, and K ⊕ . . . ⊕ K m ⊂ K ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( K ⊕ . . . ⊕ K m ) θ K ⊕ . . . ⊕ θ m K m ⊂ C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( θ K ⊕ . . . ⊕ θ m K m ) where c > is an absolute constant. The main tool is the Pr´ekopa-Leindler inequality; that is a functional formof the Brunn-Minkowski inequality. The inequality itself, due to Pr´ekopa[62] and Leindler [53] in dimension one, was generalized in Pr´ekopa [63]and [64], C. Borell [12], and in Brascamp, Lieb [16]. Various applicationsare provided and surveyed in Ball [3], Barthe [9], and Gardner [38]. Thefollowing multiplicative version from [3], is often more useful and is moreconvenient for geometric applications.8
HEOREM 3.1 (Pr´ekopa-Leindler) If λ ∈ (0 , and h, f, g are non-negative integrable functions on R n satisfying h ((1 − λ ) x + λy ) ≥ f ( x ) − λ g ( y ) λ for x, y ∈ R n , then Z R n h ≥ (cid:18)Z R n f (cid:19) − λ · (cid:18)Z R n g (cid:19) λ . (9)The case of equality in Theorem 3.1 has been characterized by Dubuc[26]. In Boroczky, De [13], the following stability version of the Prekopa-Leindler inequality is verified. THEOREM 3.2 If λ ∈ (0 , and f, g are log-concave probability densitieson R n satisfying Z R n sup z =(1 − λ ) x + λy f ( x ) − λ g ( y ) λ dz ≤ ε for ε > , then there exists w ∈ R n such that Z R n | f ( x ) − g ( x + w ) | dx ≤ ω λ ( ε ) (10) where ω λ ( ε ) = c n n n (cid:16) ε min { λ, − λ } (cid:17) for some absolute constant c > . THEOREM 3.3 If λ ∈ (0 , and unconditional convex bodies K and C in R n satisfy V ( K − λ · C λ ) ≤ (1 + ε ) V ( K ) − λ V ( C ) λ for ε > , then there exists positive definite diagonal matrix Φ such that V ( K ∆(Φ C )) < ω λ ( ε ) V ( K ) and V ((Φ − K )∆ C ) < ω λ ( ε ) V ( C ) (11) where ω λ ( ε ) is taken from (10) .Proof: To simplify notation, for any unconditional convex body L , we write L + = L ∩ R n + . We may assume that V ( K ) = V ( C ) = 1 . If ω λ ( ε ) ≥ , then we may choose Φ to be any linear map with det Φ = 1,and V ( K ∆(Φ C )) < ε > ω λ ( ε ) < . (12)9e set M = K − λ · C λ , and consider the log-concave functions f, g, h : R n → [0 , ∞ ) defined by f ( x , . . . , x n ) = K + ( e x , . . . , e x n ) e x + ... + x n g ( x , . . . , x n ) = C + ( e x , . . . , e x n ) e x + ... + x n h ( x , . . . , x n ) = M + ( e x , . . . , e x n ) e x + ... + x n . In particular, h ( z ) = sup z =(1 − λ ) x + λy f ( x ) − λ g ( y ) λ holds for any z ∈ R n by the definition of the coordinatewise product. Inaddition, Z R n h = V ( M + ) = V ( M )2 n ≤ (1 + ε ) (cid:18) V ( K )2 n (cid:19) − λ (cid:18) V ( C )2 n (cid:19) λ = (1 + ε ) (cid:18)Z R n f (cid:19) − λ (cid:18)Z R n g (cid:19) λ = 1 + ε. Therefore Theorem 3.2 yields that there exists w = ( w , . . . , w n ) ∈ R n suchthat Z R n | f ( x ) − g ( x + w ) | dx ≤ ω λ ( ε ) . Let Φ ∈ GL( n ) be the diagonal transformation Φ( t , . . . , t n ) = ( e − w t , . . . , e − w n t n );therefore, g ( x + w ) = a (Φ C ) + ( e x , . . . , e x n ) e x + ... + x n = a ˜ g ( x ) where a = e w + ... + w n . We deduce that ω λ ( ε ) V ( K + ) ≥ Z R n + | f ( x ) − a ˜ g ( x ) | dx = Z R n + | K + − a (Φ C ) + | = | a − | V ( K + ∩ ( T C ) + ) + V ( K + \ (Φ C ) + ) + aV ((Φ C ) + \ K + ) . In particular, we have V ( K + \ (Φ C ) + ) ≤ ω λ ( ε ) V ( K + ) , (13)and hence (12) implies that V ( K + ∩ (Φ C ) + ) ≥ V ( K + ). In turn, we deduce | a − | ≤ ω λ ( ε ) V ( K + ) V ( K + ∩ (Φ C ) + ) ≤ ω λ ( ε ) < , a > . It follows that V ((Φ C ) + \ K + ) ≤ ω λ ( ε ) V ( K + ) a < ω λ ( ε ) V ( K + ) . (14)Combining (13) and (14) yields V ( K + ∆(Φ C ) + ) < ω λ ( ε ) V ( K + ), and hence V ( K ∆(Φ C )) < ω λ ( ε ) V ( K ).Finally, V ( K ∆(Φ C )) < ω λ ( ε ) V ( K ) and ω λ ( ε ) ≤ yield that V (Φ C ) ≥ V ( K ), and hence V ( K ∆(Φ C )) < ω λ ( ε ) V (Φ C ). ✷ The main additional tool in this section is to strengthen the containmentrelation K − λ · C λ ⊂ (1 − λ ) · K + λ · C. We recall that e , . . . , e n form the fixed orthonormal basis of R n . For aproper subset J ⊂ { , . . . , n } , we set L J = lin { e i } i ∈ J . We observe that for a diagonal matrix T = ( t , . . . , t n ), we have k T k ∞ = max i =1 ,...,n | t i | . We write B n to denote the unit ball centered at the origin. PROPOSITION 4.1 If τ ∈ (0 , ] , λ ∈ ( τ, − τ ) , K is an unconditionalconvex body in R n and Φ is a positive definite diagonal matrix satisfying V ((1 − λ ) · K + λ · (Φ K )) ≤ (1 + ε ) V ( K − λ · (Φ K ) λ ) for ε > , then either k s Φ − I n k ∞ ≤ n · ε n τ for s = (det Φ) − n , orthere exist s , . . . , s m > and a partition of { , . . . , n } into proper subsets J , . . . , J m , m ≥ , such that m M k =1 ( L J k ∩ K ) ⊂ n · ε n τ ! K where for k = 1 , . . . , m , we have s k · ( L J k ∩ K ) ⊂ Φ( L J k ∩ K ) ⊂ n · ε n τ ! s k · ( L J k ∩ K ) . roof: First we assume that ε < τ n n n n . (15)Let Φ = ( α , . . . , α n ). We may also assume that e i ∈ ∂ Φ λ K = ∂ ( K − λ · (Φ K ) λ ) for i = 1 , . . . , n. Let θ = 8 n · ε n τ < n . We write i ⊲⊳ j for i, j ∈ { , . . . , n } ifexp( − θ ) ≤ α i α j ≤ exp( θ ) . In addition, we write ∼ to denote the the equivalence relation on { , . . . , n } induced by ⊲⊳ ; namely, for i, j ∈ { , . . . , n } , we have i ∼ j if and only ifthere exist pairwise different i , . . . , i l ∈ { , . . . , n } with i = i , i l = j , and i k − ⊲⊳ i k for k = 1 , . . . , l . We may readily assume that l ≤ n in the definition of i ∼ j. (16)Let J , . . . , J m , m ≥ ∼ . Thereason behind introducing ∼ are the estimates (17), (i) ad (ii). We claimthat if k = 1 , . . . , m and β k = min { α i : i ∈ J k } , then any x ∈ L J k satisfies β k k x k ≤ k Φ x k ≤ e n θ β k k x k . (17)To prove (17), we choose ˜ i, ˜ j ∈ J k satisfying α ˜ i = min { α i : i ∈ J k } = β k and α ˜ j = max { α i : i ∈ J k } . We deduce from (16) that α ˜ j /α ˜ i ≤ e n θ , andhence β k ≤ α i ≤ e n θ β k holds for i ∈ J k , proving (17).Next, if k = l holds for k, l ∈ { , . . . , m } , then the definition of therelation ∼ yields that either min { α i : i ∈ J k } ≥ e θ · max { α j : i ∈ J l } , ormax { α i : i ∈ J k } ≤ e − θ · min { α j : i ∈ J l } ; therefore, (i) either k Φ x kk x k ≥ e θ · k Φ y kk y k for any x ∈ L J k \ o and y ∈ L J l \ o ; (ii) or k Φ x kk x k ≤ e − θ · k Φ y kk y k for any x ∈ L J k \ o and y ∈ L J l \ o .12f m = 1, and hence F = { , . . . , n } , then (17) yields that e − nθ β − ≤ s = (det Φ) − n ≤ β − , and hence nθ ≤ k s Φ − I n k ∞ ≤ nθ. (18)Therefore we assume that m ≥
2. Here again (17) yields that if k =1 , . . . , m , then β k · ( L J k ∩ K ) ⊂ Φ( L J k ∩ K ) ⊂ (1 + 2 nθ ) β k · ( L J k ∩ K ) . (19)For M = m M k =1 ( L J k ∩ Φ λ K ) , we observe that 1 √ n B n ⊂ M ⊂ √ n B n . (20)We prove indirectly that(1 − √ nθ ) M ⊂ Φ λ K, (21)what would complete the proof of Proposition 4.1. In particular, we supposethat (1 − √ nθ ) M Φ λ K, (22)and seek a contradiction.Let η > η ( M + θB n ) ⊂ Φ λ K. We deduce that 12 n ≤ η < − √ nθ. (23)where the upper bound follows from (22), and the lower bound follows from n M ⊂ Φ λ K and the consequence θB n ⊂ M of (20).Let R ≥ = { x ∈ R : x ≥ } . The maximality of η and the uncondition-ality of K yield that there exists an x ∈ η ( M + θB n ) ∩ ∂ (Φ λ K ) ∩ R ≥ , w ∈ S n − ∩ R ≥ to ∂ (Φ λ K ) at x satisfying ( cf. (23)) x − θ n · w + θ n · B n ⊂ Φ λ K. (24)In addition, we have x + θ B n ⊂ η ( M + θB n ) + θB n ⊂ (1 − √ nθ ) M + 2 θB n ⊂ M. (25)We claim that k w | L J k k ≤ − θ n for k = 1 , . . . , m. (26)Let v ∈ S n − ∩ L J k satisfy w | L J k = k w | L J k k v , and hence k w | L J k k = h w, v i . Since k x k ≤ √ n by (20) and x − ( x | L J k ) is orthogonal to v , we have |h w, x − ( x | L J k ) i| = k x − ( x | L J k ) k p − h w, v i i ≤ √ n p − h w, v i i . It follows from (25) that ( x | L J k ) + θv ∈ K ∩ L J k . Since w is an exterior normal to K at x , we have h w, x i ≥ h w, ( x | L J k ) + θv i , thus √ n p − h w, v i ≥ h w, x − ( x | L J k ) i ≥ θ h w, v i . We deduce that k w | L J k k = h w, v i ≤ nn + θ = 1 − θ n + θ < − θ n , proving (26).In turn, we conclude from P mk =1 k w | L J k k = 1, m ≤ n and (26) thatthere exist p = q satisfying k w | L J p k ≥ θ n and k w | L J q k ≥ θ n . Possibly after reindexing, we may assume that k w | L J k ≥ θ n and k w | L J k ≥ θ n . (27)14or any u ∈ S n − ∩ R ≥ , it follows from applying first the H¨older in-equality that h u, x i ≤ h u, Φ − λ x i − λ h u, Φ − λ x i λ ≤ h K ( u ) − λ h Φ K ( u ) λ . (28)In particular, (28) implies that x ∈ (1 − λ ) · K + λ · (Φ K ).In order to prove (21); more precisely, to prove that (22) is false, ourcore statement is the following stability version of (28). CLAIM 4.2
For any u ∈ S n − ∩ R ≥ , we have h u, x i (cid:18) τ θ n . (cid:19) ≤ h K ( u ) − λ h Φ K ( u ) λ . (29) Proof:
We observe that h u, Φ − λ x i = h Φ − λ u, x i , h u, Φ − λ x i = h Φ − λ u, x i , h K ( u ) = h Φ λ K (Φ − λ u ); h Φ K ( u ) = h Φ λ K (Φ − λ u ) , and hence it follows from (28) that it is sufficient to prove that if u ∈ S n − ,then either (cid:16) h Φ λK (Φ − λ u ) h Φ − λ u,x i (cid:17) − λ ≥ τθ n . , or (cid:16) h Φ λK (Φ − λ u ) h Φ − λ u,x i (cid:17) λ ≥ τθ n . . (30)Let us write w = ⊕ mk =1 w k and u = ⊕ mk =1 u k for w k = w | L J k and u k = u | L J k ,and prove that ( cf. (27)) there exists i ∈ { , } such thateither (cid:12)(cid:12)(cid:12)(cid:12) k Φ − λ u i kk Φ − λ u k − k w i k (cid:12)(cid:12)(cid:12)(cid:12) ≥ θ n , or (cid:12)(cid:12)(cid:12)(cid:12) k Φ − λ u i kk Φ − λ u k − k w i k (cid:12)(cid:12)(cid:12)(cid:12) ≥ θ n . (31)We prove (31) by contradiction; thus, we suppose that if i ∈ { , } , then (cid:12)(cid:12)(cid:12)(cid:12) k Φ − λ u i kk Φ − λ u k − k w i k (cid:12)(cid:12)(cid:12)(cid:12) < θ n and (cid:12)(cid:12)(cid:12)(cid:12) k Φ − λ u i kk Φ − λ u k − k w i k (cid:12)(cid:12)(cid:12)(cid:12) < θ n . and seek a contradiction. Since k w k ≥ θ n and k w k ≥ θ n according to(27), we deduce that if i ∈ { , } , then e − θ < k Φ − λ u i kk Φ − λ u k · k w i k < e θ , and e − θ < k Φ − λ u i kk Φ − λ u k · k w i k < e θ . (32)It follows from Φ (cid:0) Φ − λ u (cid:1) = Φ − λ u , Φ (cid:0) Φ − λ u (cid:1) = Φ − λ u , and (32) that e − θ < k Φ (cid:0) Φ − λ u (cid:1) kk Φ − λ u k < e θ and e − θ < k Φ (cid:0) Φ − λ u (cid:1) kk Φ − λ u k < e θ ;15herefore, e θ < k Φ (cid:0) Φ − λ u (cid:1) kk Φ − λ u k : k Φ (cid:0) Φ − λ u (cid:1) kk Φ − λ u k < e θ . Since Φ − λ u i ∈ L J i for i = 1 ,
2, the last inequalities contradict (i) and (ii),and in turn verify (31).Based on (31), the triangle inequality yields the existence of i ∈ { , } such thateither (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u i k Φ − λ u k − w i (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n or (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u i k Φ − λ u k − w i (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n , and in turn we deduce thateither (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u k Φ − λ u k − w (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n or (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u k Φ − λ u k − w (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n . (33)First, we assume that out of the two possibilities in (33), we have (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u k Φ − λ u k − w (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n . (34)According to (24), we have e B = x − θ n · w + θ n · B n ⊂ Φ λ K, which in turn yields (using (34) and k x k ≤ √ n at the end) that h Φ λ K (Φ − λ u ) − h Φ − λ u, x i ≥ h e B (Φ − λ u ) − h Φ − λ u, x i = (cid:28) Φ − λ u, x − θ n · w + θ n · Φ − λ u k Φ − λ u k (cid:29) − h Φ − λ u, x i = k Φ − λ u k · θ n · (cid:28) Φ − λ u k Φ − λ u k , Φ − λ u k Φ − λ u k − w (cid:29) = k Φ − λ u k · θ n · (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u k Φ − λ u k − w (cid:13)(cid:13)(cid:13)(cid:13) ≥ h Φ − λ u, x i√ n · θ n (cid:18) θ n (cid:19) = h Φ − λ u, x i · θ n . . We conclude using 1 − λ ≥ τ that (cid:18) h Φ λ K (Φ − λ u ) h Φ − λ u, x i (cid:19) − λ ≥ (cid:18) h Φ λ K (Φ − λ u ) h Φ − λ u, x i (cid:19) τ ≥ τ θ n . . (35)16econdly, if (cid:13)(cid:13)(cid:13)(cid:13) Φ − λ u k Φ − λ u k − w (cid:13)(cid:13)(cid:13)(cid:13) ≥ θ n holds in (33), then similar argument yields (cid:18) h Φ λ K (Φ − λ u ) h Φ − λ u, x i (cid:19) λ ≥ τ θ n . . proving (30). In turn, we conclude (29) in Claim 4.2. ✷ Let ̺ ≥ x + ̺B n ⊂ (1 − λ ) · K + λ · (Φ K ) . (36)We claim that ̺ ≥ τ θ n . (37)It follows from Claim 4.2 that ̺ >
0. To prove (37), we may assume that ̺ ≤ τ θ n < n . (38)We consider a y ∈ ( x + ̺B n ) ∩ ∂ (cid:0) (1 − λ ) · K + λ · (Φ K ) (cid:1) ∩ R n ≥ , which exists as (1 − λ ) · K + λ · (Φ K ) is unconditional. Let u ∈ S n − ∩ R n ≥ be the exterior unit normal to f M = (1 − λ ) · K + λ · (Φ K )at y , and hence y = x + ̺ u . On the one hand, ± e i ∈ f M for i = 1 , . . . , n yields that h f M ( u ) ≥ √ n , thus (38) implies h u, x i = h u, y i − ̺ = h f M ( u ) − ̺ ≥ √ n . (39)On the other hand, h f M ( u ) = h K ( u ) − λ h AK ( u ) λ holds because y is a smoothboundary point of f M ; therefore, it follows from (36), (38) and (39) that ̺ = h f M ( u ) − h u, x i = h K ( u ) − λ h AK ( u ) λ − h u, x i≥ h u, x i · τ θ n . ≥ τ θ n , V (Φ λ K ) ≤ n because of ± e i ∈ (Φ λ K ), i = 1 , . . . , n , κ n = π n Γ( n +1) > ( π e ) n √ n · n n , and the supporting hyperplane at x to Φ λ K cuts x + ̺B n into half, we deduce that V ( f M ) ≥ V (Φ λ K ) + ̺ n κ n ≥ V (Φ λ K ) + κ n τ n θ n · n n n = V (Φ λ K ) (cid:18) κ n τ n θ n · n n n V (Φ λ K ) (cid:19) > V (Φ λ K ) πe ) n τ n θ n √ n · n n . n ! > V (Φ λ K ) (cid:18) τ n θ n n n n (cid:19) > (1 + ε ) V (Φ λ K ) = (1 + ε ) V ( K ) − λ V (Φ K ) λ , what is absurd. This contradicts (22), and verifies (1 − √ nθ ) M ⊂ Φ λ K ,completing the proof of Proposition 4.1 under the condition ε < τ n n n n ( cf. (15)). On the other hand, if ε ≥ τ n n n n , then16 n · ε n τ ≥ n, thus Proposition 4.1 readily holds. ✷ The proof of Theoem 2.3 will be based on Theorem 3.3 and Proposition 4.1.However, first we need some simple lemmas. The first statement is thefollowing corollary of the logarithmic Brunn-Minowski inequality for uncon-ditional convex bodies (see Lemma 3.1 of Kolesnikov, Milman [52]).
LEMMA 5.1 If K and C are unconditional convex bodies in R n , then ϕ ( t ) = V ((1 − t ) · K + t · C ) is log-concave on [0 , . The second claim provides simple estimates about log-concave functions.
LEMMA 5.2
Let ϕ be a log-concave function on [0 , . i) If λ ∈ (0 , , η ∈ (0 , · min { − λ, λ } ) and ϕ ( λ ) ≤ (1 + η ) ϕ (0) − λ ϕ (1) λ ,then ϕ (cid:0) (cid:1) ≤ (cid:18) η min { − λ, λ } (cid:19) p ϕ (0) ϕ (1) (ii) If ϕ (0) = ϕ (1) = 1 and ϕ ′ (0) ≤ , then ϕ (cid:0) (cid:1) ≤ ϕ ′ (0) .Proof: For (i), we may assume that 0 < λ < , and hence λ = (1 − λ ) · λ · , ϕ ( λ ) ≤ (1 + η ) ϕ (0) − λ ϕ (1) λ and the log-concavity of ϕ yield(1 + η ) ϕ (0) − λ ϕ (1) λ ≥ ϕ ( λ ) ≥ ϕ (0) − λ ϕ (cid:0) (cid:1) λ . Thus (1 + η ) λ ≤ e η λ ≤ ηλ implies ϕ (cid:0) (cid:1) ≤ (1 + η ) λ p ϕ (0) ϕ (1) ≤ (cid:16) ηλ (cid:17) p ϕ (0) ϕ (1) . For (ii), we write ϕ ( t ) = e W ( t ) for a concave function W with W (0) = W (1) = 0. Thus W ( ) ≤ W ′ (0), which in turn yields using W ′ (0) = ϕ ′ (0) ≤ ϕ (cid:0) (cid:1) = e W ( ) ≤ e W ′ (0) / ≤ W ′ (0) = 1 + ϕ ′ (0) . ✷ We also need the following statement about volume difference.
LEMMA 5.3 If M ⊂ K are o -symmetric convex bodies with V ( K \ M )) ≤ n +1 V ( K ) , then K ⊂ · (cid:18) V ( K \ M ) V ( M ) (cid:19) n ! M. Proof:
Let t ≥ K ⊂ (1 + t ) M. Then there exists z ∈ ∂K and y ∈ ∂M with z = (1 + t ) x , and hence22 + t · z + t t · M ⊂ K \ int M. It follows that V ( K \ M ) ≥ (cid:16) t t (cid:17) n · V ( M ), which inequality, together with V ( K \ M )) ≤ n V ( M ), implies t ≤ · (cid:16) V ( K \ M ) V ( M ) (cid:17) n . ✷ We will need the case λ = of Theorem 3.3 and Proposition 4.1.19 OROLLARY 5.4
If the unconditional convex bodies K and C in R n satisfy V ( K · C ) ≤ (1 + ε ) V ( K ) V ( C ) for ε > , then there exists positive definite diagonal matrix Φ such that V ( K ∆(Φ C )) < c n n n ε V ( K ) (40) where c > is an absolute constant. COROLLARY 5.5 If K is an unconditional convex body in R n and Φ isa positive definite diagonal matrix satisfying V (cid:18) · K + · (Φ K ) (cid:19) ≤ (1 + ε ) V ( K · (Φ K ) ) for ε > , then either k s Φ − I n k ∞ ≤ n · ε n for s = (det Φ) − n , orthere exist s , . . . , s m > and a partition of { , . . . , n } into proper subsets J , . . . , J m , m ≥ , such that m M k =1 ( L J k ∩ K ) ⊂ (cid:16) n · ε n (cid:17) Ks k ( L J k ∩ K ) ⊂ Φ( L J k ∩ K ) ⊂ (cid:16) n · ε n (cid:17) s k ( L J k ∩ K ) , k = 1 , . . . , m. Proof of Theorem 2.3
First we consider the case λ = , and hence provethat if the unconditional convex bodies K and C in R n satisfy V (cid:18) · K + · C (cid:19) ≤ (1 + ε ) V ( K ) V ( C ) (41)for ε >
0, then for m ≥
1, there exist θ , . . . , θ m > K , . . . , K m > K i , i = 1 , . . . , m , arecomplementary coordinate subspaces, and K ⊕ . . . ⊕ K m ⊂ K ⊂ (cid:16) c n ε n (cid:17) ( K ⊕ . . . ⊕ K m ) (42) θ K ⊕ . . . ⊕ θ m K m ⊂ C ⊂ (cid:16) c n ε n (cid:17) ( θ K ⊕ . . . ⊕ θ m K m ) (43)where c > ε < γ − n n − n (44)20or a suitable absolute constant γ > γ is a chosen in a way suchthat ˜ c n n n ε < n +1 (45)for the constant ˜ c of Corollary 5.4.We have V ( K · C ) ≤ V (cid:18) · K + · C (cid:19) ≤ (1 + ε ) V ( K ) V ( C ) ;therefore, Corollary 5.4 yields a positive definite diagonal matrix Φ suchthat V ((Φ K )∆ C ) < ˜ c n n n ε V ( C ) and V ( K ∆(Φ − C )) < ˜ c n n n ε V ( K ) (46)where ˜ c > M = K ∩ (Φ − C ) , and hence (46) yields that V ( M ) > (1 − ˜ c n n n ε ) V ( K ) (47) V (Φ M ) > (1 − ˜ c n n n ε ) V ( C ) . (48)As M ⊂ K and Φ M ⊂ C , it follows that V (cid:0) M + (Φ M ) (cid:1) ≤ (1 + ε ) V ( K ) V ( C ) ≤ (1 + 2˜ c n n n ε ) V ( M ) V (Φ M ) = (1 + 2˜ c n n n ε ) V ( M · (Φ M ) ) . Now we apply Corollary 5.5, and deduce the existence of an absolute con-stant c > k s Φ − I n k ∞ ≤ c n · ε n for s = (det Φ) − n , orthere exist s , . . . , s m > { , . . . , n } into proper subsets J , . . . , J m , m ≥
2, such that m M k =1 ( L J k ∩ M ) ⊂ (cid:16) c n · ε n (cid:17) M where for k = 1 , . . . , m , we have s k · ( L J k ∩ M ) ⊂ Φ( L J k ∩ M ) ⊂ (cid:16) c n · ε n (cid:17) s k · ( L J k ∩ M ) .
21e deduce from (45), (47), (48), and Lemma 5.3 the existence of an absoluteconstant c > M ⊂ K ⊂ (1 + c nε n ) M Φ M ⊂ C ⊂ (1 + c nε n )Φ M. Now if k s Φ − I n k ∞ ≤ c n · ε n , then we can choose m = 1 and K = M to verify Theorem 2.3. On the other and, if k s Φ − I n k ∞ l > c n · ε n , thenwe choose K k = (cid:16) c n · ε n (cid:17) − ( L J k ∩ M ) for k = 1 , . . . , m. For c = c + c + c c and c = c + c + c c , it follows that m M k =1 K k ⊂ M ⊂ K ⊂ (1 + c nε n ) M ⊂ (1 + c nε n ) m M k =1 ( L J k ∩ M ) ⊂ (cid:16) c n · ε n (cid:17) m M k =1 K km M k =1 s k K k ⊂ m M k =1 Φ K k ⊂ Φ M ⊂ C ⊂ (1 + c nε n )Φ M. ⊂ (1 + c nε n ) m M k =1 Φ( L J k ∩ M ) ⊂ (1 + c nε n ) m M k =1 (cid:16) c n · ε n (cid:17) s k · ( L J k ∩ M ) ⊂ (cid:16) c n · ε n (cid:17) m M k =1 s k ( L J k ∩ M ) ⊂ (cid:16) c n · ε n (cid:17) m M k =1 s k K k . This proves Theorem 2.3 if λ = and ε < γ − n n − n ( cf. (44)).Still keeping λ = , we observe that if Q is any unconditional convexbody in R n , then n M i =1 ( R e i ∩ Q ) ⊂ nQ. (49)Therefore, if ε ≥ γ − n n − n ( cf. (44)) holds in (41), then (42) and (43)readily hold for suitable absolute constant c > m = n , K k = n ( R e k ∩ K ), and choosing θ k > θ k ( R e k ∩ K ) = R e k ∩ C k = 1 , . . . , n . In particular, Theorem 2.3 has been verified if λ = .Next, we assume that λ ∈ [ τ, − τ ] holds for some τ ∈ (0 , ] in Theo-rem 2.3. First let ε ≤ τ . Since ϕ ( t ) = V ((1 − t ) · K + t · C )is log-concave on [0 ,
1] according to Lemma 5.1, Lemma 5.2 yields that ϕ (cid:0) (cid:1) ≤ (cid:18) ε min { − λ, λ } (cid:19) p ϕ (0) ϕ (1);or in other words, V (cid:18) · K + · C (cid:19) ≤ (cid:16) ετ (cid:17) V ( K ) V ( C ) . We deduce from (42) and (43) that for m ≥
1, there exist θ , . . . , θ m > K , . . . , K m > K i , i = 1 , . . . , m , are complementary coordinate subspaces, and K ⊕ . . . ⊕ K m ⊂ K ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( K ⊕ . . . ⊕ K m ) (50) θ K ⊕ . . . ⊕ θ m K m ⊂ C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( θ K ⊕ . . . ⊕ θ m K m ) . (51)Finally, if λ ∈ [ τ, − τ ] holds for some τ ∈ (0 , ] in Theorem 2.3 and ε ≥ τ , then choosing again m = n , K k = n ( R e k ∩ K ), and θ k > θ k ( R e k ∩ K ) = R e k ∩ C for k = 1 , . . . , n , (49) yields (50) and (51). ✷ In this section, we verify some properties of the part of a convex bodycontained in a Weyl chamber.
LEMMA 6.1
Let H , . . . , H n be independent linear ( n − -dimensionalsubspaces, and let W be the closure of a connected component of R n \ ( H ∪ . . . ∪ H n ) . i) If M is a convex body in R n symmetric through H , . . . , H n , then ν M,q ∈ W for any q ∈ W ∩ ∂ ′ M , and in turn M ∩ W = { x ∈ W : h x, u i ≤ h M ( u ) ∀ u ∈ W } . (ii) If λ ∈ (0 , and K and C are convex bodies in R n symmetric through H , . . . , H n , then W ∩ ((1 − λ ) K + λ C ) = { x ∈ W : h x, u i ≤ h K ( u ) − λ h C ( u ) λ ∀ u ∈ W } . Proof:
For (i), it is sufficient to prove the first statement; namely, if q ∈ int W ∩ ∂ ′ K , then ν M,q ∈ W .Let u i ∈ S n − , i = 1 , . . . , n , such that W = { x ∈ R n : h x, u i i ≥ } , andhence h q, u i i > i = 1 , . . . , n , and (i) is equivalent with the statement thatif i = 1 , . . . , n , then h u i , ν K,q i ≥ . (52)Since q ′ = q − h q, u i i u i is the reflexted image of q through H i , we have q ′ ∈ M ; therefore,0 ≤ h ν K,q , q − q ′ i = h ν K,q , h q, u i i u i i = 2 h q, u i i · h ν K,q , u i i . As h q, u i i >
0, we conclude (52), and in turn (i).For (ii), let M = (1 − λ ) K + λ C , and let M + = { x ∈ W : h x, u i ≤ h K ( u ) − λ h C ( u ) λ ∀ u ∈ W } . Readily, W ∩ M ⊂ M + . Therefore, (ii) follows if for any q ∈ ∂ ′ M ∩ int W ,we have q ∈ ∂M + . As q ∈ ∂M ∩ int W , there exists u ∈ S n − such that h q, u i = h K ( u ) − λ h C ( u ) λ . Since q ∈ ∂ ′ M ∩ W , we have u = ν M,q , and hence(i) yields that ν M,q ∈ W . Therefore q ∈ ∂M + , proving Lemma 6.1 (ii). ✷ The main idea in order to use the known results about unconditionalconvex bodies is to linearly transfer a Weyl chamber W into the corner R n ≥ . LEMMA 6.2
Let K be a convex body in R n with o ∈ int K , let independent v , . . . , v n ∈ R n satisfy that h v i , v j i ≥ for ≤ i ≤ j ≤ n , let W =pos { v , . . . , v n } , and let Φ W = R n ≥ for a Φ ∈ GL ( n, R ) .(i) Φ − t W ⊂ R n ≥ .(ii) If ν K,x ∈ W for any x ∈ W ∩ ∂ ′ K , then ν Φ K,z ∈ R n ≥ for any z ∈ R n ≥ ∩ ∂ ′ Φ K ; (53)24 iii) and there exists an unconditional convex body K such that R n ≥ ∩ K = Φ( W ∩ K ) . Proof:
Let e , . . . , e n be the standard orthonormal basis of R n indexed in away such that e i = Φ v i . First we claim that h e i , Φ − t v i ≥ v ∈ W and i = 1 , . . . , n . (54)Since v = P nj =1 λ j v j for λ , . . . , λ n ≥
0, we deduce from h v j , v i i ≥ ≤ * n X j =1 λ j v j , v i + = h v, v i i = h Φ − t v, Φ v i i = h Φ − t v, e i i , proving (54). In turn, we deduce (i) from (54).If z ∈ W ∩ ∂ ′ K , then ν K,z ∈ W and Φ − t ν K,z is an exterior normal toΦ K at Φ z , therefore, (ii) follows from (i).Now (53) yields that if z = ( z , . . . , z n ) ∈ R n ≥ ∩ ∂ ′ Φ K and 0 ≤ y i ≤ z i , i = 1 , . . . , n , then y = ( y , . . . , y n ) ∈ Φ K . Therefore repeatedly reflecting R n ≥ ∩ Φ K through the coordinate hyperplanes, we obtain the unconditionalconvex body K such that R n ≥ ∩ K = R n ≥ ∩ Φ K = Φ( W ∩ K ). ✷ Since if a linear map A leaves a convex body K invariant, then the minimalvolume Loewner ellipsoid is also invariant under A , Barthe, Fradelizi [7]prove that it is sufficient to consider orthogonal reflections in our setting. LEMMA 7.1 (Barthe, Fradelizi)
If the convex bodies K and C in R n are invariant under linear reflections A , . . . , A n through n independent lin-ear ( n − -planes H , . . . , H n , then there exists B ∈ SL( n ) such that BA B − , . . . , BA n B − are orthogonal reflections through BH , . . . , BH n and leave BK and BC in-variant. For the theory of Coxeter groups, we follow Humpreys [47]. For an n -dimensional real vector space V equipped with a Euclidean structure, let G be closure of the Coxeter group generated by the orthogonal reflectionsthrough p ⊥ , . . . , p ⊥ n for independent p , . . . , p n ∈ V . A linear subspace L of V is invariant under G if and only if p , . . . , p n ∈ L ∪ L ⊥ . We say that25n invariant linear subspace L is irreducible if L = { o } and any invariantsubspace L ′ ⊂ L satisfies either L ′ = L or L ′ = { o } , and hence the actionof G on an irreducible invariant subspace is irreducible. Since the intersec-tion and the orthogonal complement of invariant subspaces is invariant, theirreducible subspaces L , . . . , L m , m ≥ L ⊕ . . . ⊕ L m = V. (55)It follows that any A ∈ G can be written as A = A | L ⊕ . . . ⊕ A | L m . For aninvariant subspace L ⊂ V , we set G | L = { A | L : A ∈ G } , and write O ( L )to denote the group of isometries of L fixing the origin. In particular, ourmain task is to understand irreducible Coxeter groups. LEMMA 7.2 (Barthe, Fradelizi)
Let G be closure of the Coxeter groupgenerated by the orthogonal reflections through p ⊥ , . . . , p ⊥ n for independent p , . . . , p n ∈ R n . If L ⊂ R n is an irreducible invariant subspace, and G | L isinfinite, then G | L = O ( L ) . Next, if L is an irreducible invariant d -dimensional linear subspace of V with repect to the closure G of a Coxeter group and G | L is finite, thena more detailed analysis is needed. To set up the correponding notation,let G ′ = G | L be the finite Coxeter group generated by some orthogonalreflections acting on L . Let H , . . . , H k ⊂ L be the linear ( d − G ′ are the ones through H , . . . , H k ,and let u , . . . , u k ∈ L \{ o } be a system of roots for G ′ ; namely, there areexactly two roots orthogonal to each H i , and these two roots are opposite.We note that for algebraic purposes, one usually normalize the roots in away such that h u i ,u j i h u i ,u i i is an integer but we drop this condition because weare only interested in the cones determined by the roots.Let W be the closure of a Weyl chamber; namely, a connected componentof L \{ H , . . . , H k } . It is known (see [47]) that W = pos { v , . . . , v d } = ( d X i =1 λ i v i : ∀ λ i ≥ ) where v , . . . , v d ∈ L are independent. In addition, for any x ∈ L \{ H , . . . , H k } ,there exists a unique A ∈ G ′ such that x ∈ AW , and hence the Weyl cham-bers are in a natural bijective correspondence with G ′ . We may reindex H , . . . , H k and u , . . . , u k in a way such that H i = u ⊥ i for i = 1 , . . . , d arethe ”walls” of W , and h u i , v i i > i = 1 , . . . , d ; h u i , v j i = 0 for 1 ≤ i < j ≤ d . (56)26n this case, reflections L → L through H , . . . , H d generate G ′ , and u , . . . , u d is called a simple system of roots. The order we list simple roots is not re-lated to the corresponding Dynkin diagram. LEMMA 7.3
Let G be the Coxeter group generated by the orthogonal re-flections through p ⊥ , . . . , p ⊥ n for independent p , . . . , p n ∈ R n . If L ⊂ R n is an irreducible invariant d -dimensional subspace with d ≥ , and G | L isfinite, and W = pos { v , . . . , v d } ⊂ L is the closure of a Weyl chamber for G | L , then h v i , v j i ≥ d · k v i k · k v j k . (57) Proof:
Let G ′ = G | L . We use the classification of finite irreducible Coxetergroups. For the cases when G ′ is either of D d , E , E , E (see Adams [1]about E , E , E ), we use the known simple systems of roots in terms of theorthonormalt basis e , . . . , e d of L to construct v , . . . , v d via (56). However,there is a unified construction for the other finite irreducible Coxeter groupsbecause they are the symmetries of some regular polytopes. Case 1: G ′ is one of the types I ( m ), A d , B d , F , H , H In this case, G ′ is the symmetry group of some d -dimensional regular poly-tope P centered at the origin. Let F ⊂ . . . ⊂ F d − be a tower of facesof P where dim F i = i , i = 0 , . . . , d −
1. Defining v i to be the centroid of F i − , i = 1 , . . . , d , we have that W = pos { v , . . . , v d } is the closure of aWeyl chamber because the symmetry group of P is simply transitive on thetowers of faces of P .As G ′ is irreducible, the John ellipsoid of P (the unique ellipsoid oflargest volume contained in P ) is a d -dimensional ball centered at the originof some radius r >
0. It follows that P ⊂ drB n , and hence r ≤ k v i k ≤ dr for i = 1 , . . . , d . In addition, v i is the closest point of aff F i − to the originfor i = 1 , . . . , d , and v j ∈ F i − if 1 ≤ j ≤ i , thus h v j , v i i = h v i , v i i if1 ≤ j ≤ i ≤ d . We conclude that if 1 ≤ j ≤ i ≤ d , then h v j , v i ik v j k · k v i k = k v i kk v j k ≥ d . Case 2: G ′ = D n In this case, a simple system of roots is u i = e i − e i +1 for i = 1 , . . . , d − ,u d = e d − + e d .
27n turn, we may choose v , . . . , v d as v i = P il =1 e l for i = 1 , . . . , d − i = d,v d − = − v d + P d − l =1 e l . As h v i , v j i is a positive integer for i = j , and k v i k ≤ √ d for i = 1 , . . . , d , weconclude (57). Case 3: G ′ = E In this case d = 6, and a simple system of roots is u i = e i − e i +1 for i = 1 , , , ,u = e + e u = √ e − P l =1 e l . Using coordinates in e , . . . , e , we may choose v , . . . , v as v = ( √ , , , , , v = ( √ , √ , , , , v = ( √ , √ , √ , , , v = (1 , , , , − , √ v = (1 , , , , , √ ) and v = (0 , , , , , h v i , v j i ≥ i = j , and k v i k ≤ √
18 for i = 1 , . . . ,
6, we conclude (57).
Case 4: G ′ = E In this case d = 7, and a simple system of roots is u i = e i − e i +1 for i = 1 , , , , ,u = e + e u = √ e − P l =1 e l . Using coordinates in e , . . . , e , we may choose v , . . . , v as v = (2 , , , , , , √ v = (1 , , , , , , √ v = (1 , , , , , , √ ), v = (1 , , , , , , √ v = (1 , , , , , − , √ v = (1 , , , , , , √
2) and v = (0 , , , , , h v i , v j i ≥ i = j , and k v i k < √
28 for i = 1 , . . . ,
7, we conclude (57).
Case 5: G ′ = E In this case d = 8, and a simple system of roots is u i = e i − e i +1 for i = 1 , , , , , , ,u = − P l =1 e l + P l =6 e l . Using coordinates in e , . . . , e , we may choose v , . . . , v as v = (1 , − , − , − , − , − , − , − v = (0 , , − , − , − , − , − , − v = ( − , − , − , − , − , − , − , − v = ( − , − , − , − , − , − , − , − v = ( − , − , − , − , − , − , − , − ),28 = ( − , − , − , − , − , − , − , − v = ( − , − , − , − , − , − , − , − v = ( − , − , − , − , − , − , − , − h v i , v j i ≥ i = j , and k v i k < √
48 for i = 1 , . . . ,
8, we conclude (57). ✷ For a convex body invariant under a Coxeter group, we can determinethe some exterior normal at certain points provided by the symmetries ofthe convex body.
LEMMA 7.4
Let G be the closure of a Coxeter group generated by n in-dependent orthogonal reflections of R n , let L ⊂ R n be an irreducible linearsubspace and let K be a convex body in R n invariant under G .(i) If G | L is finite, and W = pos { v , . . . , v d } ⊂ L is the closure of a Weylchamber for G | L , and t i v i ∈ ∂K for t i > , i = 1 , ldots, d , then v i isan exterior normal at tv i .(ii) If G | L is infinite and v ∈ L \{ o } , and tv ∈ ∂K for t > , then v is anexterior normal at tv .Proof: Let d = dim L .For (i), first we claim that there exist independent u , . . . , u n − ∈ v ⊥ i suchthat the reflection through u ⊥ j lies in G for j = 1 , . . . , n −
1. To construct u , . . . , u n − ∈ v ⊥ i , if d ≥
2, then we choose roots u , . . . , u d − ∈ v ⊥ i for G | L that corresponds to the walls of W containing v i . In addition, if d < n , thenwe choose independent u d , . . . , u n − ∈ L ⊥ such that the reflection through u ⊥ j lies in G for j = d, . . . , n −
1, completing the construction of u , . . . , u n − .Let N = { z ∈ R n : h z, t i v i − x i ≥ ∀ x ∈ K } be the normal cone at t i v i ∈ ∂K . If N = R ≥ v i , then we are done. Since N is a cone and o ∈ int K ,if N = R ≥ v i , then there exists w ∈ v ⊥ i \{ o } such that z = v i + w ∈ N . Let H ⊂ G be the closure of the subgroup generated by the reflections through u ⊥ , . . . , u ⊥ n − , and hence both R v i and v ⊥ i are invariant under H . Since u , . . . , u n − ∈ v ⊥ i are independent, the centroid of M = conv { Aw : A ∈ H } ⊂ v ⊥ i is o . We deduce that the centroid of v i + M = conv { Aw : A ∈ H } ⊂ N is v i ; therefore, v i ∈ N .For (ii), the argument is essentially same because similarly, there existindependent ˜ u , . . . , ˜ u n − ∈ v ⊥ such that the reflection through ˜ u ⊥ j lies in G for j = 1 , . . . , n − ✷ The proof Theorem 1.4
Lemma 7.1 and the linear invariance of the L -sum yield that we may assumethat A , . . . , A n are orthogonal reflections through the linear ( n − H , . . . , H n , respectively, with H ∩ . . . ∩ H n = { o } where K and C areinvariant under A , . . . , A n .Let G be the closure of the group generated by A , . . . , A n , and let L , . . . , L m be the irreducible invariant subspaces of R n of the action of G . If t , . . . , t m > ∈ GL( n, R ) satisfies Ψ x = t i x for x ∈ L i and i = 1 , . . . , m , thenΨ K and Ψ C are both invariant under G . (58)Let E be the John ellipsoid of K , that is, the unique ellipsoid of maximalvolume contained in K . Therefore, E is also invariant under G . In partic-ular, we can choose the principal directions of E in a way such that eachis contained in one of the L i , and L i ∩ E is a Euclidean ball of dimensiondim L i . Therefore, after applying a suitable linear transformation like in(58), we may assume that E = B n , and hence B n ⊂ K ⊂ nB n . (59)For any i = 1 , . . . , n , let G i = G | L i if G | L i is finite, and let G i be thesymmetry group of some dim L i dimensional regular simplex in L i centeredat the origin if G | L i is infinite.We consider the finite subgroup e G ⊂ G that is the direct sum of G , . . . , G m ,acting in the natural way e G | L i = G i for i = 1 , . . . , m . Let 0 = p < p <. . . < p m = n satisfy that p i − p i − = dim L i for i = 1 , . . . , m . We choosea basis v , . . . , v n ∈ S n − of R n , in a way such that for each i = 1 , . . . , m , W i = pos { v p i − +1 , . . . , v p i } is the closure of a Weyl chamber for the irre-ducible action of G i on L i .According to Lemma 7.3, these v , . . . , v n ∈ S n − satisfy that h v j , v l i ≥ n if p i − + 1 ≤ j < l ≤ p i and i = 1 , . . . , m ; (60) h v j , v l i = 0 if there exists i = 1 , . . . , m − j ≤ p i < l .(61)Let e , . . . , e n be the standard orthonormal basis of R n , let Φ ∈ GL( n ) satisfythat Φ v i = e i , i = 1 , . . . , n , and let W = W ⊕ . . . ⊕ W m .
30t follows that Φ W = R n ≥ and int W is a fundamental domain for e G in thesense that S { AW : A ∈ e G } = R n int AW ∩ int BW = ∅ if A, B ∈ e G and A = B . (62)If i ∈ { , . . . , m } and p i − + 1 ≤ j ≤ p i , then we define u j ∈ L i ∩ S n − by h u j , v j i > h u j , v l i = 0 for l = j . Therefore, u ⊥ , . . . , u ⊥ n are thewalls of W ; namely, the linear hulls of the facest of the simplicial cone W ,and the reflections through u ⊥ , . . . , u ⊥ n are symmetries of both K and C (and actually generate e G ). We may apply Lemma 6.2 to W because ofLemma 6.1, (60) and (61), and deduce the existence unconditional convexbodies e K and e C such that R n ≥ ∩ e K = Φ( W ∩ K ) and R n ≥ ∩ e C = Φ( W ∩ C ) . We claim that R n ≥ ∩ ((1 − λ ) e K + λ e C ) ⊂ Φ ( W ∩ ((1 − λ ) K + λC )) . (63)According to Lemma 6.1 and to Φ − t W ⊂ R n ≥ ( cf. Lemma 6.2), we have R n ≥ ∩ ((1 − λ ) e K + λ e C ) = { x ∈ R n ≥ : h x, u i ≤ h e K ( u ) − λ h e C ( u ) − λ ∀ u ∈ R n ≥ }⊂ { x ∈ R n ≥ : h x, u i ≤ h e K ( u ) − λ h e C ( u ) λ ∀ u ∈ Φ − t W } . We observe that if u ∈ Φ − t W , then there exist y ∈ R n ≥ ∩ ∂ e K = R n ≥ ∩ ∂ (Φ K )and z ∈ R n ≥ ∩ ∂ e C = R n ≥ ∩ ∂ (Φ C ) with h e K ( u ) = h y , u i and h e C ( u ) = h z , u i .For v = Φ t u ∈ W , y = Φ − y ∈ W ∩ ∂K and y = Φ − y ∈ W ∩ ∂K , itfollows that v is an exterior normal to K at y and to C at z , and h e K ( u ) − λ h e C ( u ) λ = h Φ y, Φ − t v i − λ h Φ z, Φ − t v i λ = h y, v i − λ h z, v i λ = h K ( v ) − λ h C ( v ) λ . We deduce from the considerations just above and from applying Lemma 6.1to W that R n ≥ ∩ ((1 − λ ) e K + λ e C ) ⊂ Φ { q ∈ W : h q, v i ≤ h K ( v ) − λ h K ( v ) λ ∀ v ∈ W } = Φ ( W ∩ ((1 − λ ) K + λC )) , proving (63).Writing | e G | to denote the cardinality of e G , (62) yields V ( M ) = | e G | · V ( M ∩ W )31here M is either K , C or (1 − λ ) · K + λ · C . We deduce from (63) andthe condition in Theorem 1.4 that V ((1 − λ ) · e K + λ · e C ) = 2 n V (cid:16) R n ≥ ∩ ((1 − λ ) · e K + λ · e C ) (cid:17) ≤ n V (Φ ( W ∩ ((1 − λ ) K + λC )))= 2 n | det Φ || e G | · V ((1 − λ ) K + λC ) ≤ n | det Φ || e G | · (1 + ε ) V ( K ) − λ V ( C ) λ = (1 + ε ) V ( e K ) − λ V ( e C ) λ . We apply the following equivalent form of Theorem 2.3 to e K and e C where λ ∈ [ τ, − τ ] for τ ∈ (0 , ]. There exist absolute constant ˜ c >
1, comple-mentary coordinate linear subspaces e Λ , . . . , e Λ k , k ≥
1, with ⊕ kj =1 e Λ j = R n such that ⊕ kj =1 (cid:16) e K ∩ e Λ j (cid:17) ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) e K, (64)and there exist θ , . . . , θ k > ⊕ kj =1 θ j (cid:16) e K ∩ e Λ j (cid:17) ⊂ e C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ⊕ kj =1 θ j (cid:16) e K ∩ e Λ j (cid:17) . (65)For Λ j = Φ − e Λ j , j = 1 , . . . , k , we deduce that W ∩ k X j =1 ( K ∩ Λ j ) ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( W ∩ K ) , (66)and W ∩ k X j =1 θ j ( K ∩ Λ j ) ⊂ W ∩ C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) W ∩ k X j =1 θ j ( K ∩ Λ j ) . (67)We observe that each Λ j is spanned by a subset of v , . . . , v n .For the rest of the argument, first we assume that ε is small enough tosatisfy ˜ c n (cid:16) ετ (cid:17) n < n . (68)We claim that if (68) holds, theneach Λ j , j = 1 , . . . , k , is invariant under G . (69)32e suppose indirectly that the claim (69) does not hold, and we seek acontradiction. In this case, k ≥
2. Since each Λ j is spanned by a subset of v , . . . , v n , after possibly reindexing L , . . . , L m , Λ , . . . , Λ k and v , . . . , v n ,we may assume that v ∈ L ∩ Λ and v ∈ L ∩ Λ . For i = 1 , . . . , n , let s i > s i v i ∈ ∂K ; therefore, (59) yields1 ≤ s i ≤ n, (70)and hence s v ∈ L ∩ K ∩ Λ and v ∈ L ∩ K ∩ Λ . (71)It follows from (60) that h v , v i ≥ n . (72)We deduce from (71), and then from (66) that s v + v ∈ W ∩ k X j =1 ( K ∩ Λ j ) ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ( W ∩ K ) . (73)Lemma 7.4 yields that v is an exterior unit normal to ∂K at s v , andhence s = h K ( v ). We deduce from first (73) and then from assumption(68) and the formula (70) that s + h v , v i = h v , s v + v i ≤ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) h K ( v )= s + ˜ c n (cid:16) ετ (cid:17) n s < n . (74)On the other hand, we have s + h v , v i ≥ n by (72), contradicting (74).In turn, we conclude (69) under the assumption (68).We deduce from (66), (67), (69) and the symmetries of K and C that ⊕ kj =1 ( K ∩ Λ j ) ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) K, (75)and ⊕ kj =1 θ j ( K ∩ Λ j ) ⊂ C ⊂ (cid:18) c n (cid:16) ετ (cid:17) n (cid:19) ⊕ kj =1 θ j ( K ∩ Λ j ) . (76)In addition, the symmetries of K and (69) yield that K ∩ Λ j = K | Λ j for j = 1 , . . . , k , therefore, K ⊂ ⊕ kj =1 ( K ∩ Λ j ) . c n (cid:16) ετ (cid:17) n ≥ n , (77)and hence (4˜ c ) n (cid:16) ετ (cid:17) n ≥ n . (78)For i = 1 , . . . , m , the symmetries of K and C yield that r i ( B n ∩ L i ) isthe John ellipsoid of K ∩ L i and θ i r i ( B n ∩ L i ) is the John ellipsoid of C ∩ L i for some r i , θ i >
0. For K i = r i n ( B n ∩ L i ), i = 1 , . . . , m , we have ⊕ mi =1 K i ⊂ conv { mK , . . . , mK m } ;therefore, it follows from (78) that ⊕ mi =1 K i ⊂ K ⊂ n · ⊕ mi =1 K i ⊂ (cid:18) c ) n (cid:16) ετ (cid:17) n (cid:19) ⊕ mi =1 K i ⊕ mi =1 θ i K i ⊂ C ⊂ n · ⊕ mi =1 θ i K i ⊂ (cid:18) c ) n (cid:16) ετ (cid:17) n (cid:19) ⊕ mi =1 θ i K i , proving Theorem 1.4 under the assumption (77). ✷ As in the case of Theorem 1.4, it follows from Lemma 7.1 and the linearinvariance of the L -sum that we may assume that A , . . . , A n are orthogonalreflections through the linear ( n − H , . . . , H n , respectively, with H ∩ . . . ∩ H n = { o } where K and C are invariant under A , . . . , A n . Wewrite G to denote the closure of the group generated by A , . . . , A n , and L , . . . , L m to denote the irreducible invariant subspaces of R n of the actionof G .For the logarithmic Minkowski Conjecture 1.2, replacing either K or C by a dilate does not change the difference of the two sides; therefore, wemay assume that V ( K ) = V ( C ) = 1 . In this case, the condition in Theorem 1.5 states that Z S n − log h C h K dV K < ε (79)34or ε > nε < , (80)for t ∈ [0 , ϕ ( t ) = V ((1 − t ) · K + t · C ) . According to (3.7) in B¨or¨oczky, Lutwak, Yang, Zhang [15], we have ϕ ′ (0) = n Z S n − log h C h K dV K , (81)and hence (79) and the assumption (80) yield that ϕ ′ (0) < nε where nε < V (cid:18) · K + · C (cid:19) = ϕ (cid:18) (cid:19) < nε. Now we apply Theorem 1.4, and conclude that for some m ≥
1, there exist θ , . . . , θ m > K , . . . , K m > G such that lin K i , i = 1 , . . . , m , are complementary coordinate subspaces,and K ⊕ . . . ⊕ K m ⊂ K ⊂ (cid:16) c n ε n (cid:17) ( K ⊕ . . . ⊕ K m ) (82) θ K ⊕ . . . ⊕ θ m K m ⊂ C ⊂ (cid:16) c n ε n (cid:17) ( θ K ⊕ . . . ⊕ θ m K m ) (83)where c > nε < nε ≥
1, then Theorem 1.5 can be proved as Theo-rem 1.4 under the assumption (77). ✷ Acknowledgement
We would like to thank Gaoyong Zhang and RichardGardner for illuminating discussions.
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