Stability of the Logarithmic Brunn-Minkowski Inequality

arXiv:2101.02549v3 [math.MG] 10 Mar 2021 ayl[0 1,Kn 6] ade,Zaic 4] Eskenaz [49], Zvavitch new Gardner, developing rapidly [62], The t Kane [37]. in s 61], work setting, [60, related Gaussian recent the Sanyal For and theory [85]). algorithmic Vu valuations, Tao, and ( [83] geometry v Schneider convex of and core combinatorics the additive form probability, inequality Brunn-Minkowski classical The Introduction 1 MSC aet ugr,[email protected] Hungary, dapest, stecasclcs)iiitdb uwk[8 9 0,hsb has 70], 69, [68, Lutwak by initiated case) classical the is ay n eateto ahmtc,CnrlErpa Univ [email protected] European Hungary, Central Budapest, Mathematics, of Department and gary, ∗ † ‡ eiae oPo.EwnLta nteocso fhsseventy his of occasion the on Lutwak Erwin Prof. to Dedicated lre ńiIsiueo ahmtc,Ratnd .13- u. Realtanoda 132002 Mathematics, K Alfréd of Rényi grant Institute NKFIH by Supported eateto ahmtc,CnrlErpa nvriy N University, European Central Mathematics, of Department tblt fteLgrtmcBrunn-Minkowski Logarithmic the of Stability nqaiyadteLgrtmcMnosiieult o ovxbodies convex established. for inequality are Brunn-Minkows Minkowski Logarithmic Logarithmic the the and of inequality versions stability a hyperplanes, 52A40 nqaiyi h aeo ayhyperplane many of case the in inequality ntecs fsmere ihrsetto respect with symmetries of case the In ´ rl .BöröczkyKároly J. ac 1 2021 11, March symmetries Abstract L birthday p BunMnositer (where theory -Brunn-Minkowski 1 pai De Apratim , ∗ † n 5 -03Bdps,Hun- Budapest, H-1053 15, needn linear independent riy ao .9 H-1051, 9, u. Nador ersity, ‡ dru ,H15,Bu- H-1051, 9, u. ador esyJochemko, say ee e ade [48], Gardner see rosaesin areas arious s Moschidis is, eter of theory he cm main ecome -fifth ki p 1 = research area in modern convex geometry and geometric analysis. Following Firey [45] and Lutwak [68, 69, 70], major results have been obtained by Hug, Lutwak, Yang, Zhang [58], and more recently the papers Kolesnikov, Milman [65], Chen, Huang, Li, Liu [26], Hosle, Kolesnikov, Livshyts [57], Kolesnikov, Livshyts [64] present new developments and approaches. We note that the Lp-Minkowski and Lp-Brunn-Minkowski inequalities are even extended to certain families of non-convex sets by Zhang [90], Ludwig, Xiao, Zhang [67] and Lutwak, Yang, Zhang [71]. We call a compact compact set K in Rn a convex body if V (K) > 0 where V (K) stands for the n-dimensional Lebesgue measure. The corner- stone of the Brunn-Minkowski Theory is the Brunn-Minkowski inequality (see Schneider [83]). If K and C are convex bodies in Rn and α,β > 0, then the Brunn-Minkowski inequality says that

1 1 1 V (αK + βC) n αV (K) n + βV (C) n (1) ≥ where equality holds if and only if C = γK + z for γ > 0 and z Rn. ∈ Because of the homogeneity of the Lebesgue measure, (1) is equivalent to say that if λ (0, 1), then ∈ 1 1 λ λ V ((1 λ)K + λβC) n V (K) − V (C) (2) − ≥ where equality holds if and only if K and C are translates. We also note another consequence of the Brunn-Minkowski inequality (1); namely, the Minkowski inequality says that

hC dSK hK dSK provided V (C)= V (K). (3) n−1 ≥ n−1 ZS ZS The first stability forms of the Brunn-Minkowski inequality were due to Minkowski himself (see Groemer [52]). If the distance of K and C is measured in terms of the so-called Hausdorff distance, then Diskant [33] and Groemer [51] provided close to be optimal stability versions (see Groemer [52]). However, the natural distance is in terms volume of the symmetric difference, and the optimal result is due to Figalli, Maggi, Pratelli [41, 42]. To define the “homothetic distance” A(K,C) of convex bodies K and C, let −1 −1 α = K n and β = C n , and let | | | | A(K,C) = min αK∆(x + βC) : x Rn {| | ∈ } where K∆Q stands for the symmetric difference of K and Q. In addition, C K let σ(K,C) = max |K| , |C | . Now Figalli, Maggi, Pratelli [42] proved that | | | | n o 2 n−1 3 (2 2 n ) 2 2 setting γ∗ = ( −122n7 ) , we have

1 1 1 γ∗ n n n 2 K + C ( K + C ) 1+ 1 A(K,C) . | | ≥ | | | | " σ(K,C) n · # Here the exponent 2 of A(K,C)2 is optimal (cf. Figalli, Maggi, Pratelli [42]). We note that prior to [42], the only known error term in the Brunn- Minkowski inequality was of order A(K,C)η with η n, due to Diskant ≥ [33] and Groemer [51] in their work on providing stability result in terms of the Hausdorff distance (see also Groemer [52]), and also to a more direct approach by Esposito, Fusco, Trombetti [38]; therefore, the exponent depended significantly on n. We note that recently, various breakthrough stability results about geometric functional inequalities have been obtained. Fusco, Maggi, Pratelli [47] proved an optimal stability version of the isoperimetric inequality (whose result was extended to the Brunn-Minkowski inequality by Figalli, Maggi, Pratelli [41, 42], see also Eldan, Klartag [36]). Stonger versions of the Borell- Brascamp-Lieb inequality are provided by Ghilli, Salani [50] and Rossi, Salani [80], and of the Sobolev inequality by Figalli, Zhang [44] (extend- ing Bianchi, Egnell [13] and Figalli, Neumayer [43]), Nguyen [75] and Wang [89], and of some related inequalities by Caglar, Werner [23]. Related inequalities are verified by Colesanti [29], Colesanti, Livshyts, Marsiglietti [30], P. Nayar, T. Tkocz [73, 74], Xi, Leng [86]. In this paper, we focus on the L0 sum of replacing Minkowski addition. First, for λ (0, 1), the L or logarithmic sum of two origin symmetric ∈ 0 convex bodies K and C in Rn is defined by

n 1 λ λ n 1 (1 λ) K + λ C = x R : x, u h (u) − h (u) u S − . − · 0 · ∈ h i≤ K C ∀ ∈ n o It is linearly invariant, as A((1 λ) K+ λ C) = (1 λ) A K+ λ AC for A − · 0 · − · 0 · ∈ GL(n). The following strengthening of the Brunn-Minkowski inequality for centered convex bodies is a long-standing and highly investigated conjecture.

CONJECTURE 1.1 (Logarithmic Brunn-Minkowski conjecture) If λ (0, 1) and K and C are convex bodies in Rn whose centroid is the origin, ∈ then 1 λ λ V ((1 λ) K + λ C) V (K) − V (C) , (4) − · 0 · ≥ with equality if and only if K = K1 + . . . + Km and C = C1 + . . . + Cm compact convex sets K1,...,Km,C1,...,Cm of dimension at least one where m i=1 dim Ki = n and Ki and Ci are dilates, i = 1,...,m. P 3 We note that the choice of the right translates of K and C are important in Conjecture 1.1 according to the examples by Nayar, Tkocz [73]. On the other hand, the following is an equivalent form of the origin symmetric case of the Logarithmic Brunn-Minkowski conjecture for o-symmetric convex bodies. n 1 The cone volume measure or L0-surface area measure VK on S − , whose study was initiated independently by Firey [46] and Gromov and Milman [53], has become an indispensable tool in the last decades (see say Barthe, Guédon, Mendelson, Naor [12], Naor [72], Paouris, Werner [76]). If a convex body K contains the origin, then its cone volume measure is 1 dVK = n hK dSK where hK is the support function of K and the total measure is the volume of K. Following partial and related results by Andrews [2], Chou, Wang [28], He, Leng, Li [54], Henk, Schürman, Wills [56], Stancu [84], Xiong [87] the paper Boroczky, Lutwak, Yang, Zhang [21] characterized even cone volume measures by the so called subspace concentration condition. Recently, breakthrough results have been obtained by Chen, Li, Zhu [27], Chen, Huang, Li [26], Kolesnikov [63], Nayar, Tkocz [74], Kolesnikov, Milman [65], Putter- man [79] about the uniqueness of the solution, which is intimately related to the conjectured log-Minkowski inequality Conjecture 1.2. As it turns out, subspace concentration condition also holds for the cone-volume measure VK if the centroid of a general convex body K is the origin (see Henk, Linke [55] and Böröczky, Henk [17, 18]).

CONJECTURE 1.2 (Logarithmic Minkowski conjecture) If K and C are convex bodies in Rn whose centroid is the origin, then

hC V (K) V (C) log dVK log n−1 h ≥ n V (K) ZS K with the same equality conditions as in Conjecture 1.1.

Actually understanding the equality case in the Logarithmic Minkowski Conjecture 1.2 for o-symmetric convex bodies clariﬁes the uniqueness of the solution of the Monge-Ampere type logarithmic Minkowski Problem in the even case (see Boroczky, Lutwak, Yang, Zhang [20], Kolesnikov, Milman [65], Chen, Huang, Li, Liu [26]). In R2, Conjecture 1.1 is veriﬁed in Boroczky, Lutwak, Yang, Zhang [20] for o-symmetric convex bodies, but it is still open in general. On the other hand, Xi, Leng [86] proved that any two dimensional convex bodies K and C in R2 can be translated in a way such that (4) holds for the translates.

4 In higher dimensions, Conjecture 1.1 is proved for with enough hyperplane symmetries (cf. Theorem 1.3) and complex bodies (cf. Rotem [81]). For o-symmetric convex bodies, Conjecture 1.2 is proved when K is close to be an ellipsoid by a combination of the local estimates by Kolesnikov, Mil- man [65] and the use of the continuity method in PDE by Chen, Huang, Li, Liu [26]. Another even more recent proof of this result based on Alexan- drov’s approach of considering the Hilbert-Brunn-Minkowski operator for polytopes is due to Putterman [79]. Additional local versions of Conjec- ture 1.2 for o-symmetric convex bodies are due to Kolesnikov, Livshyts [64]. We say that A GL(n) is a linear reflection associated to the linear ∈ (n 1)-space H Rn if A fixes the points of H and det A = 1. In this − ⊂ − case, there exists u Rn H such that Au = u where the invariant subspace ∈ \ − Ru is uniquely determined (see Davis [32], Humphreys [59], Vinberg [88]). It follows that a linear reflection A is a classical ”orthogonal” reflection if and only if A O(n). ∈ Following the result on unconditional convex bodies by Saroglou [82], Boroczky, Kalantzopoulos [19] verified the logarithmic Brunn-Minkowski and Minkowski conjectures under hyperplane symmetry assumption. THEOREM 1.3 (Boroczky, Kalantzopoulos) If the convex bodies K n and C in R are invariant under linear reflections A1,...,An through n hyperplanes H ,...,H with H . . . H = o , then 1 n 1 ∩ ∩ n { } 1 λ λ V ((1 λ) K + λ C) V (K) − V (C) (5) − · 0 · ≥ hC V (K) V (C) log dVK log , (6) n−1 h ≥ n V (K) ZS K with equality in either inequality if and only if K = K1 + . . . + Km and C = C1 + . . . + Cm for compact convex sets K1,...,Km,C1,...,Cm of dimension at least one and invariant under A1,...,An where Ki and Ci are dilates, m i = 1,...,m, and i=1 dim Ki = n. Geometric inequalitiesP under n independent hyperplane symmetries were first considered by Barthe, Fradelizi [11] and Barthe, Cordero-Erausquin [10]. These papers verified the classical Mahler conjecture and Slicing conjecture, respectively, for these type of bodies. The main result of our paper is a stability version of Theorem 1.3. 1 THEOREM 1.4 If λ [τ, 1 τ] for τ (0, 2 ], the convex bodies K and n ∈ − ∈ C in R are invariant under linear reflections A1,...,An through n hyperplanes H ,...,H with H . . . H = o , and 1 n 1 ∩ ∩ n { } 1 λ λ V ((1 λ) K + λ C) (1 + ε)V (K) − V (C) − · 0 · ≤

5 for ε> 0, then for some m 1, there exist compact convex sets K ,C ,...,K ,C ≥ 1 1 m m of dimension at least one and invariant under A1,...,An where Ki and Ci m are dilates, i = 1,...,m, and i=1 dim Ki = n such that

1 P ε 95n K + . . . + K K 1+ cn (K + . . . + K ) 1 m ⊂ ⊂ τ 1 m 1 ε 95n C + . . . + C C 1+ cn (C + . . . + C ) 1 m ⊂ ⊂ τ 1 m where c> 1 is an absolute constant.

Let us present an example showing that the bound of Theorem 1.4 is not far from being optimal in the sense that the exponent 1/(95n) should be at least 1/n. If for small ε > 0, K is obtained from the box K0 = 1 1 1 n 1 n [ 2−n−1 , 2n−1 ] [ 2, 2] − by cutting oﬀ corners of size of order ε , and C is × − n 1 n 1 1 1 n 1 obtained from the box C0 = [ 2 − , 2 − ] [ −2 , 2 ] − by cutting oﬀ corners 1 − 1 ×1 n of suitable size of order ε n , then K + C = [ 1, 1] , and 2 · 0 2 · − 1 1 1 1 V K + C (1 + ε)V (K) 2 V (C) 2 , 2 · 0 2 · ≤ 1 but if ηK K for η> 0, then η 1 γ ε n where γ > 0 depends on n. 0 ⊂ ≤ − We deduce from Theorem 1.4 a stability version of the logarithmic- Minkowski inequality (6) for convex bodies with many hyperplane symmetries.

THEOREM 1.5 If the convex bodies K and C in Rn are invariant under linear reﬂections A ,...,A through n hyperplanes H ,...,H with H 1 n 1 n 1 ∩ . . . H = o , and ∩ n { } h dV 1 V (C) log C K log + ε n−1 h V (K) ≤ n · V (K) ZS K for ε> 0, then for some m 1, there exist compact convex sets K ,C ,...,K ,C ≥ 1 1 m m of dimension at least one and invariant under A1,...,An where Ki and Ci m are dilates, i = 1,...,m, and i=1 dim Ki = n such that

P n 1 K + . . . + K K 1+ c ε 95n (K + . . . + K ) 1 m ⊂ ⊂ 1 m n 1 C + . . . + C C 1+ c ε 95n (C + . . . + C ) 1 m ⊂ ⊂ 1 m where c> 1 is an absolute constant.

6 To prove Theorem 1.4, first we verify it in the unconditional case, see Section 2 presenting these partial results. More precisely, first we consider the coordinatewise product of unconditional convex bodies based on the recent stability version of the Prekopa-Leindler inequality (see Section 3), and then handle the unconditional case Theorem 2.3 of Theorem 1.4 in Sections 4 and 5. Next we review some fundamental properties of Weyl chambers and Coxeter groups in general in Section 6 and Section 7, and prove Theorem 1.4 in Section 8. Finally, Theorem 1.5 is verified in Section 9.

2 The case of unconditional convex bodies

The way to prove Theorem 1.4 is ﬁrst clarifying the case of unconditional convex bodies; namely, when A1,...,An are orthogonal reﬂections and H1,...,Hn are coordinate hyperplanes. For unconditional convex bodies, the coordinatewise product is a classical tool; namely, if λ (0, 1) and K and C are ∈ unconditional convex bodies in Rn, then

1 λ λ 1 λ λ 1 λ λ n K − C = ( x − y ,..., x − y ) R : · { ±| 1| | 1| ±| n| | n| ∈ (x ,...,x ) K and (y ,...,y ) C . 1 n ∈ 1 n ∈ } It is known that (see say Saroglou [82]) that K1 λ Cλ is a convex uncon- − · ditional body, and it follows from the Hölder inequality (see also Saroglou [82]) that 1 λ λ K − C (1 λ) K + λ C. · ⊂ − · 0 · In addition, [82] verifies that if λ (0, 1), T is a positive definite diagonal ∈ matrix and K is an unconditional convex body in Rn, then

1 λ λ λ K − (T K) = T K (7) · η η η where T = (t ,...,tn) for η R and T = (t ,...,t ) for t ,...,t > 0. 1 ∈ 1 n 1 n The Logarithmic Brunn-Minkowski Conjecture 1.1 was veriﬁed for unconditional convex bodies by several authors, as Bollobas, Leader [14] and Cordero-Erausquin, Fradelizi, Maurey [31] veriﬁed the inequality V ((1 λ) − · K + λ C) V (K)1 λV (C)λ in (8) about the coordinatewise product, 0 · ≥ − even before the log-Brunn-Minkowski conjecture was stated, and the containment relation between the coordinatewise product and the L0-sum and the description of the equality case are due to Saroglou [82]. For X,Y Rn, ⊂ we write X Y to denote X + Y if linX and linY are orthogonal. ⊕

7 THEOREM 2.1 (Saroglou) If K and C are unconditional convex bodies in Rn and λ (0, 1), then ∈ 1 λ λ 1 λ λ V ((1 λ) K + λ C) V (K − C ) V (K) − V (C) . (8) − · 0 · ≥ · ≥ (i) V (K1 λ Cλ) = V (K)1 λV (C)λ if and only if C = ΦK for a positive − · − definite diagonal matrix Φ. (ii) V ((1 λ) K + λ C) = V (K)1 λV (C)λ if and only if K = K − · 0 · − 1 ⊕ . . . K and L = L . . . L for unconditional compact convex ⊕ m 1 ⊕ ⊕ m sets K1,...,Km,L1,...,Lm of dimension at least one where Ki and Li are dilates, i = 1,...,m. We note that the second inequality in (8) (about the coordinatewise product) is a consequence of the Prekopa-Leindler inequality (see Section 3). In turn, the stability version Proposition 3.2 of the Prekopa-Leindler inequality yields the following: THEOREM 2.2 If λ [τ, 1 τ] for τ (0, 1 ], and the unconditional ∈ − ∈ 2 convex bodies K and C in Rn satisfy 1 λ λ 1 λ λ V (K − C ) (1 + ε)V (K) − V (C) · ≤ for ε> 0, then there exists positive definite diagonal matrix Φ such that 1 1 n n ε 19 1 n n ε 19 V (K∆(ΦC)) < c n V (K) and V ((Φ− K)∆C) < c n V (C) τ τ where c> 1 is an absolute constant. In the case of the logarithmic-Brunn-Minkowski inequality for unconditional convex bodies, we have a different type stability estimate: THEOREM 2.3 If λ [τ, 1 τ] for τ (0, 1 ], and the unconditional ∈ − ∈ 2 convex bodies K and C in Rn satisfy 1 λ λ V ((1 λ) K + λ C) (1 + ε)V (K) − V (C) − · 0 · ≤ for ε > 0, then for some m 1, there exist θ ,...,θ > 0 and uncondi- ≥ 1 m tional compact convex sets K1,...,Km such that lin Ki, i = 1,...,m, are complementary coordinate subspaces, and 1 ε 95n K . . . K K 1+ cn (K . . . K ) 1 ⊕ ⊕ m ⊂ ⊂ τ 1 ⊕ ⊕ m 1 ε 95n θ K . . . θ K C 1+ cn (θ K . . . θ K ) 1 1 ⊕ ⊕ m m ⊂ ⊂ τ 1 1 ⊕ ⊕ m m where c> 1 is an absolute constant.

8 3 Coordinatewise product

The main tool is the Prékopa-Leindler inequality; that is, a functional form of the Brunn-Minkowski inequality, due to Prékopa [77] and Leindler [66] in dimension one, and to Prékopa [78], C. Borell [15] and Brascamp, Lieb [22] in higher dimensions (see Artstein-Avidan, Florentin, Segal [4] for a recent variant). Various applications are provided and surveyed in Ball [5], Barthe [9] and Gardner [48]. The following multiplicative version from [5] is the most convenient for geometric applications. THEOREM 3.1 (Prékopa-Leindler) If λ (0, 1) and h, f, g are non- ∈ negative integrable functions on Rn satisfying h((1 λ)x+λy) f(x)1 λg(y)λ − ≥ − for x,y Rn, then ∈ 1 λ λ − h f g . (9) Rn ≥ Rn · Rn Z Z Z The case of equality in Theorem 3.1 has been characterized by Dubuc [34], and the functions f, g and h should be essentially log-concave in the case of equality. Here a non-negative function ϕ on Rn is log-concave if ϕ((1 λ)x + λy) ϕ(x)1 λϕ(y)λ for all x,y Rn and λ (0, 1). In − ≥ − ∈ ∈ Boroczky, De [16], the following stability version of the Prekopa-Leindler inequality for log-concave functions is verified. THEOREM 3.2 If λ (0, 1) and f, g are log-concave probability densities ∈ on Rn satisfying

1 λ λ sup f(x) − g(y) dz 1+ ε Rn z=(1 λ)x+λy ≤ Z − for ε> 0, then there exists w Rn such that ∈

f(x) g(x + w) dx ωλ(ε) (10) Rn | − | ≤ Z 1 n n ε 19 where ωλ(ε)= c n min λ,1 λ for some absolute constant c> 1. { − } THEOREM 3.3 If λ (0, 1) and unconditional convex bodies K and C ∈ in Rn satisfy 1 λ λ 1 λ λ V (K − C ) (1 + ε)V (K) − V (C) · ≤ for ε> 0, then there exists positive deﬁnite diagonal matrix Φ such that 1 V (K∆(ΦC)) < 8ωλ(ε)V (K) and V ((Φ− K)∆C) < 12ωλ(ε)V (C) (11) where ωλ(ε) is taken from (10).

9 Proof: To simplify notation, for any unconditional convex body L, we write

L = L Rn . + ∩ + We may assume that V (K)= V (C) = 1. If ω (ε) 1 , then we may choose Φ to be any linear map with det Φ = 1, λ ≥ 4 and V (K∆(ΦC)) < 2 implies (11). Therefore, we may also assume that ε> 0 is small enough to ensure 1 ω (ε) < . (12) λ 4

1 λ λ We set M = K − C , and consider the log-concave functions f,g,h : Rn · x1 xn x1+...+xn [0, ) deﬁned by f(x1,...,xn)= 1K+ (e ,...,e )e , g(x1,...,xn)= →x1 ∞ xn x1+...+xn x1 xn x1+...+xn 1C+ (e ,...,e )e and h(x1,...,xn)= 1M+ (e ,...,e )e . In particular, 1 λ λ h(z)= sup f(x) − g(y) z=(1 λ)x+λy − holds for any z Rn by the deﬁnition of the coordinatewise product. In ∈ addition,

1 λ λ V (M) V (K) − V (C) h = V (M+)= n (1 + ε) n n Rn 2 ≤ 2 2 Z 1 λ λ − = (1+ ε) f g =1+ ε. Rn Rn Z Z Therefore Theorem 3.2 yields that there exists w = (w , . . . , w ) Rn such 1 n ∈ that

f(x) g(x + w) dx ωλ(ε). Rn | − | ≤ Z Let Φ GL(n) be the diagonal transformation Φ(t ,...,t ) = (e w1 t ,...,e wn t ); ∈ 1 n − 1 − n therefore,

x1 xn x1+...+xn w1+...+wn g(x + w)= a1(ΦC)+ (e ,...,e )e = ag˜(x) where a = e .

We deduce that

ωλ(ε)V (K+) f(x) ag˜(x) dx = 1K+ a1(ΦC)+ ≥ Rn | − | Rn | − | Z + Z + = a 1 V (K (TC) )+ V (K (ΦC) )+ aV ((ΦC) K ). | − | + ∩ + +\ + +\ +

10 In particular, we have

V (K (ΦC) ) ω (ε)V (K ), (13) +\ + ≤ λ + and hence (12) implies that V (K (ΦC) ) 3 V (K ). In turn, we deduce + ∩ + ≥ 4 + ω (ε)V (K ) 4 1 a 1 λ + ω (ε) < , | − |≤ V (K (ΦC) ) ≤ 3 λ 3 + ∩ + 2 thus a> 3 . It follows that ω (ε)V (K ) 3 V ((ΦC) K ) λ + < ω (ε)V (K ). (14) +\ + ≤ a 2 λ +

Combining (13) and (14) yields V (K+∆(ΦC)+) < 3ωλ(ε)V (K+), and hence V (K∆(ΦC)) < 3ωλ(ε)V (K). 1 Finally, V (K∆(ΦC)) < 3ωλ(ε)V (K) and ωλ(ε) 4 yield that V (ΦC) 1 ≤✷ ≥ 4 V (K), and hence V (K∆(ΦC)) < 12ωλ(ε)V (ΦC).

4 Linear images of unconditional convex bodies

The main additional tool in this section is to strengthen the containment relation 1 λ λ K − C (1 λ) K + λ C. · ⊂ − · 0 · n We recall that e1,...,en form the ﬁxed orthonormal basis of R . For a proper subset J 1,...,n , we set ⊂ { }

LJ = lin ei i J . { } ∈

We observe that for a diagonal matrix T = (t1,...,tn), we have

T = max ti . k k∞ i=1,...,n | | We write Bn to denote the unit ball centered at the origin.

PROPOSITION 4.1 If τ (0, 1 ], λ (τ, 1 τ), K is an unconditional ∈ 2 ∈ − convex body in Rn and Φ is a positive deﬁnite diagonal matrix satisfying

1 λ λ V ((1 λ) K + λ (ΦK)) (1 + ε)V (K − (ΦK) ) − · 0 · ≤ ·

11 1 − 4 ε 5n 1 for ε > 0, then either sΦ In 16n 1 for s = (det Φ) n , or k − k∞ ≤ · τ 5 there exist s ,...,s > 0 and a partition of 1,...,n into proper subsets 1 m { } J ,...,J , m 2, such that 1 m ≥ m 1 5 4 ε n (LJk K) 1 + 16n 1 K ∩ ⊂ · τ 5 ! Mk=1 where for k = 1,...,m, we have

1 5 4 ε n sk (LJk K) Φ(LJk K) 1 + 16n 1 sk (LJk K). · ∩ ⊂ ∩ ⊂ · τ 5 ! · ∩

Proof: First we assume that τ n ε< . (15) 220nn15n

Let Φ = (α1,...,αn). We may also assume that

λ 1 λ λ e ∂Φ K = ∂(K − (ΦK) ) for i = 1,...,n. i ∈ · Let 1 5 2 ε n 1 θ = 8n 1 < . · τ 5 2n We write i ⊲⊳ j for i, j 1,...,n if ∈ { } α exp( θ) i exp(θ). − ≤ αj ≤

In addition, we write to denote the the equivalence relation on 1,...,n ∼ { } induced by ⊲⊳; namely, for i, j 1,...,n , we have i j if and only if ∈ { } ∼ there exist pairwise different i ,...,i 1,...,n with i = i, i = j, and 0 l ∈ { } 0 l ik 1 ⊲⊳ ik for k = 1,...,l. We may readily assume that − l n in the definition of i j. (16) ≤ ∼ Let J ,...,J , m 1 be the equivalence classes with respect to . The 1 m ≥ ∼ reason behind introducing are the estimates (17), (i) ad (ii). We claim ∼ that if k = 1,...,m and β = min α : i J , then any x L satisfies k { i ∈ k} ∈ Jk β x Φx enθβ x . (17) kk k ≤ k k≤ kk k

12 To prove (17), we choose ˜i, ˜j J satisfying α = min α : i J = β ∈ k ˜i { i ∈ k} k and α = max α : i J . We deduce from (16) that α /α enθ, and ˜j { i ∈ k} ˜j ˜i ≤ hence β α enθβ holds for i J , proving (17). k ≤ i ≤ k ∈ k Next, if k = l holds for k, l 1,...,m , then the deﬁnition of the 6 ∈ { } relation yields that either min α : i J eθ max α : i J , or ∼ { i ∈ k} ≥ · { j ∈ l} max α : i J e θ min α : i J ; therefore, { i ∈ k}≤ − · { j ∈ l} Φx θ Φy (i) either k x k e k y k for any x LJk o and y LJl o; k k ≥ · k k ∈ \ ∈ \ Φx θ Φy (ii) or k x k e− k y k for any x LJk o and y LJl o. k k ≤ · k k ∈ \ ∈ \ If m = 1, and hence F = 1,...,n , then (17) yields that 1 { } − nθ 1 1 1 e− β− s = (det Φ) n β− , 1 ≤ ≤ 1 and hence nθ 1 implies ≤

sΦ In 2nθ. (18) k − k∞ ≤ Therefore we assume that m 2. Here again (17) yields that if k = ≥ 1,...,m, then

β (L K) Φ(L K) (1 + 2nθ)β (L K). (19) k · Jk ∩ ⊂ Jk ∩ ⊂ k · Jk ∩ For m M = (L ΦλK), Jk ∩ Mk=1 we observe that 1 Bn M √nBn. (20) √n ⊂ ⊂ We prove indirectly that

(1 2√nθ)M ΦλK, (21) − ⊂ what would complete the proof of Proposition 4.1. In particular, we suppose that (1 2√nθ)M ΦλK, (22) − 6⊂ and seek a contradiction. Let η > 0 be maximal such that

η(M + θBn) ΦλK. ⊂

13 We deduce that 1 η < 1 2√nθ. (23) 2n ≤ − where the upper bound follows from (22), and the lower bound follows from 1 M ΦλK and the consequence θBn M of (20). n ⊂ ⊂ Let R 0 = x R : x 0 . The maximality of η and the uncondition- ≥ { ∈ ≥ } ality of K yield that there exists an

n λ x0 η(M + θB ) ∂(Φ K) R 0, ∈ ∩ ∩ ≥ n 1 λ and there exists a unique exterior normal w S − R 0 to ∂(Φ K) at x0 ∈ ∩ ≥ satisfying (cf. (23))

θ θ x w + Bn ΦλK. (24) 0 − 2n · 2n · ⊂ In addition, we have

(x L )+ θv K L . 0| Jk ∈ ∩ Jk Since w is an exterior normal to K at x , we have w,x w, (x L )+ 0 h 0i ≥ h 0| Jk θv , thus i √n 1 w, v 2 w,x (x L ) θ w, v . − h i ≥ h 0 − 0| Jk i≥ h i We deduce that p n θ2 θ2 w L 2 = w, v 2 = 1 < 1 , k | Jk k h i ≤ n + θ2 − n + θ2 − 2n

14 proving (26). In turn, we conclude from m w L 2 = 1, m n and (26) that k=1 k | Jk k ≤ there exist p = q satisfying 6 P θ2 θ2 w L and w L . k | Jp k≥ 2n2 k | Jq k≥ 2n2 Possibly after reindexing, we may assume that θ2 θ2 w L and w L . (27) k | J1 k≥ 2n2 k | J2 k≥ 2n2 n 1 For any u S − R 0, it follows from applying ﬁrst the H¨older in- ∈ ∩ ≥ equality that

λ 1 λ 1 λ λ 1 λ λ u, x u, Φ− x − u, Φ − x h (u) − h (u) . (28) h 0i ≤ h 0i h 0i ≤ K ΦK In particular, (28) implies that x (1 λ) K + λ (ΦK). 0 ∈ − · 0 · In order to prove (21); more precisely, to prove that (22) is false, our core statement is the following stability version of (28).

n 1 CLAIM 4.2 For any u S − R 0, we have ∈ ∩ ≥ 5 τθ 1 λ λ u, x 1+ h (u) − h (u) . (29) h 0i 1024n5.5 ≤ K ΦK Proof: We observe that u, Φ λx = Φ λu, x , u, Φ1 λx = Φ1 λu, x , h − 0i h − 0i h − 0i h − 0i λ hK (u) = hΦλK(Φ− u); 1 λ hΦK (u) = hΦλK(Φ − u), and hence it follows from (28) that it is suﬃcient to prove that if u Sn 1, ∈ − then either

− − λ 1 λ 5 1 λ λ 5 hΦλK (Φ u) − τθ hΦλK (Φ u) τθ − 1+ 5.5 , or 1− 1+ 5.5 . (30) Φ λu,x0 1024n Φ λu,x0 1024n h i ≥ h i ≥ Let us write w = m w and u = m u for w = w L and u = u L , ⊕k=1 k ⊕k=1 k k | Jk k | Jk and prove that (cf. (27)) there exists i 1, 2 such that ∈ { } Φ λu θ2 Φ1 λu θ2 either k − ik w , or k − ik w . (31) Φ λu − k ik ≥ 16n2 Φ1 λu − k ik ≥ 16n2 − − k k k k We prove (31) by contradiction; thus, we suppose that if i 1, 2 , then ∈ { } Φ λu θ2 Φ1 λu θ2 k − ik w < and k − ik w < . Φ λu − k ik 16n2 Φ1 λu − k ik 16n2 − − k k k k

15 θ2 θ2 and seek a contradiction. Since w 2 and w 2 according to k 1k ≥ 2n k 2k ≥ 2n (27), we deduce that if i 1, 2 , then ∈ { } λ 1 λ θ Φ− ui θ θ Φ − ui θ e− 4 < k k < e 4 , and e− 4 < k k < e 4 . (32) Φ λu w Φ1 λu w k − k · k ik k − k · k ik λ 1 λ λ 1 λ It follows from Φ Φ− u1 =Φ − u1,Φ Φ− u2 =Φ − u2, and (32) that λ λ θ Φ Φ− u1 θ θ Φ Φ− u2 θ e− 2 < k k < e 2 and e− 2 < k k < e 2 ; Φ λu Φ λu k − 1k k − 2k therefore, λ λ Φ Φ− u1 Φ Φ− u2 eθ < k k : k k < eθ. Φ λu Φ λu k − 1k k − 2k Since Φ λu L for i = 1, 2, the last inequalities contradict (i) and (ii), − i ∈ Ji and in turn verify (31). Based on (31), the triangle inequality yields the existence of i 1, 2 ∈ { } such that Φ λu θ2 Φ1 λu θ2 either − i w or − i w , Φ λu − i ≥ 16n2 Φ1 λu − i ≥ 16n2 − − k k k k and in turn we deduce that

Φ λu θ2 Φ1 λu θ2 either − w or − w . (33) Φ λu − ≥ 16n2 Φ1 λu − ≥ 16n2 − − k k k k First, we assume that out of the two possibilities in (33), we have

Φ λu θ2 − w . (34) Φ λu − ≥ 16n2 − k k According to (24), we have

θ θ B = x w + Bn ΦλK, 0 − 2n · 2n · ⊂ which in turn yields (using (34) and x √n at the end) that e k 0k≤ λ λ λ λ h λ (Φ− u) Φ− u, x h e(Φ− u) Φ− u, x Φ K − h 0i ≥ B − h 0i 1 λ λ θ θ Φ − u λ = Φ− u, x w + Φ− u, x 0 − 2n · 2n · Φ1 λu − h 0i k − k 1 λ 2 1 λ θ Φ − u = Φ − u w k k · 4n · Φ1 λu − k − k λ 2 2 λ 5 Φ− u, x0 θ θ Φ − u, x0 θ h i = h i · . ≥ √n · 4n 16n2 1024n5.5 16 We conclude using 1 λ τ that − ≥ λ 1 λ λ τ 5 h λ (Φ u) − h λ (Φ u) τθ Φ K − Φ K − 1+ . (35) Φ λu, x ≥ Φ λu, x ≥ 1024n5.5 h − 0i h − 0i Secondly, if Φ1 λu θ2 − w Φ1 λu − ≥ 16n2 − k k holds in (33), then similar argument yields

1 λ λ 5 h λ (Φ u) τθ Φ K − 1+ . Φ1 λu, x ≥ 1024n5.5 h − 0i proving (30). In turn, we conclude (29) in Claim 4.2. ✷

Let ̺ 0 be maximal with the property that ≥ x + ̺Bn (1 λ) K + λ (ΦK). (36) 0 ⊂ − · 0 · We claim that τθ5 ̺ . (37) ≥ 2048n6 It follows from Claim 4.2 that ̺> 0. To prove (37), we may assume that

τθ5 1 ̺ < . (38) ≤ 2048n6 2n We consider a

n Rn y0 (x0 + ̺B ) ∂ (1 λ) K +0 λ (ΦK) 0, ∈ ∩ − · · ∩ ≥ which exists as (1 λ) K + λ (ΦK) is unconditional. Let u Sn 1 Rn − · 0 · ∈ − ∩ 0 be the exterior unit normal to ≥

M = (1 λ) K + λ (ΦK) − · 0 · at y0, and hence y0 = xf0 + ̺ u. On the one hand, ei M for i = 1,...,n 1 ± ∈ yields that hf(u) , thus (38) implies M ≥ √n f 1 u, x = u, y ̺ = hf(u) ̺ . (39) h 0i h 0i− M − ≥ 2√n

17 1 λ λ f On the other hand, hM (u)= hK (u) − hAK (u) holds because y0 is a smooth boundary point of M; therefore, it follows from (36), (38) and (39) that 1 λ λ ̺ = hf(u) u, x = h (u) − h (u) u, x fM − h 0i K AK − h 0i τθ5 τθ5 u, x , ≥ h 0i · 1024n5.5 ≥ 2048n6 proving (37). λ n λ Since V (Φ K) 2 because of ei (Φ K), i = 1,...,n, κn = n n ≤ ± ∈ π 2 (πe) 2 λ n n > n , and the supporting hyperplane at x0 to Φ K cuts x0+̺B Γ( 2 +1) 4√n n 2 into half, we deduce· that ̺nκ κ τ nθ5n V (M) V (ΦλK)+ n V (ΦλK)+ n ≥ 2 ≥ 2 2048nn6n κ τ nθ5n · f = V (ΦλK) 1+ n 2 2048nn6nV (ΦλK) · n (πe) 2 τ nθ5n τ nθ5n > V (ΦλK) 1+ > V (ΦλK) 1+ 8√n 4096nn6.5n 215nn10n · ! λ 1 λ λ > (1 + ε)V (Φ K)=(1+ ε)V (K) − V (ΦK) , what is absurd. This contradicts (22), and veriﬁes (1 2√nθ)M ΦλK, − τ⊂n completing the proof of Proposition 4.1 under the condition ε< 220nn15n (cf. τ n (15)). On the other hand, if ε 20 15 , then ≥ 2 nn n 1 5 4 ε n 16n 1 n, · τ 5 ≥ thus Proposition 4.1 readily holds. ✷

5 Proof of Theorem 2.3

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 the following corollary of the logarithmic Brunn-Minowski inequality for unconditional convex bodies (see Lemma 3.1 of Kolesnikov, Milman [65]). LEMMA 5.1 If K and C are unconditional convex bodies in Rn, then ϕ(t)= V ((1 t) K + t C) − · 0 · is log-concave on [0, 1].

18 The second claim provides simple estimates about log-concave functions.

LEMMA 5.2 Let ϕ be a log-concave function on [0, 1].

(i) If λ (0, 1), η (0, 2 min 1 λ, λ ) and ϕ(λ) (1 + η)ϕ(0)1 λϕ(1)λ, ∈ ∈ · { − } ≤ − then η ϕ 1 1+ ϕ(0)ϕ(1) 2 ≤ min 1 λ, λ { − } p (ii) If ϕ(0) = ϕ(1) = 1 and ϕ (0) 2, then ϕ 1 1+ ϕ (0). ′ ≤ 2 ≤ ′ Proof: For (i), we may assume that 0 <λ< 1 , and hence λ = (1 2λ) 0+ 2 − · 2λ 1 , ϕ(λ) (1 + η)ϕ(0)1 λϕ(1)λ and the log-concavity of ϕ yield · 2 ≤ − 1 λ λ 1 2λ 1 2λ (1 + η)ϕ(0) − ϕ(1) ϕ(λ) ϕ(0) − ϕ . ≥ ≥ 2 1 η η Thus (1 + η) 2λ e 2λ 1+ implies ≤ ≤ λ 1 1 η ϕ (1 + η) 2λ ϕ(0)ϕ(1) 1+ ϕ(0)ϕ(1). 2 ≤ ≤ λ p p For (ii), we write ϕ(t) = eW (t) for a concave function W with W (0) = W (1) = 0. Thus W ( 1 ) 1 W (0), which in turn yields using W (0) = 2 ≤ 2 ′ ′ ϕ (0) 2 that ′ ≤ ′ 1 W ( 1 ) W (0)/2 ϕ = e 2 e 1+ W ′(0) = 1 + ϕ′(0). ✷ 2 ≤ ≤ We also need the following statement about volume diﬀerence.

LEMMA 5.3 If M K are o-symmetric convex bodies with V (K M)) 1 ⊂ \ ≤ 2n+1 V (K), then

1 V (K M) n K 1 + 4 \ M. ⊂ · V (M) ! Proof: Let t 0 be minimal with ≥ K (1 + t)M. ⊂ Then there exists z ∂K and y ∂M with z = (1+ t)x, and hence ∈ ∈ 2 t z + M K int M. 2+ t · 2+ t · ⊂ \

19 n t It follows that V (K M) 2+t V (M), which inequality, together with \ ≥ · 1 1 V (K M) n V (K M)) V (M), implies t 4 \ . ✷ \ ≤ 2n ≤ · V (M) 1 We will need the case λ = 2 of Theorem 3.3 and Proposition 4.1. COROLLARY 5.4 If the unconditional convex bodies K and C in Rn satisfy 1 1 1 1 V (K 2 C 2 ) (1 + ε)V (K) 2 V (C) 2 · ≤ for ε> 0, then there exists positive deﬁnite diagonal matrix Φ such that

1 V (K∆(ΦC)) < cnnnε 19 V (K) (40) where c> 1 is an absolute constant.

COROLLARY 5.5 If K is an unconditional convex body in Rn and Φ is a positive deﬁnite diagonal matrix satisfying

1 1 1 1 V K + (ΦK) (1 + ε)V (K 2 (ΦK) 2 ) 2 · 0 2 · ≤ · − 4 1 1 for ε > 0, then either sΦ In 20n ε 5n for s = (det Φ) n , or k − k∞ ≤ · there exist s ,...,s > 0 and a partition of 1,...,n into proper subsets 1 m { } J ,...,J , m 2, such that 1 m ≥ m 4 1 (L K) 1 + 20n ε 5n K Jk ∩ ⊂ · Mk=1 4 1 s (L K) Φ(L K) 1 + 20n ε 5n s (L K), k = 1,...,m. k Jk ∩ ⊂ Jk ∩ ⊂ · k Jk ∩ 1 Proof of Theorem 2.3 First we consider the case λ = 2 , and hence prove that if the unconditional convex bodies K and C in Rn satisfy

1 1 1 1 V K + C (1 + ε)V (K) 2 V (C) 2 (41) 2 · 0 2 · ≤ for ε > 0, then for m 1, there exist θ ,...,θ > 0 and unconditional ≥ 1 m compact convex sets K1,...,Km > 0 such that lin Ki, i = 1,...,m, are complementary coordinate subspaces, and

n 1 K . . . K K 1+ c ε 95n (K . . . K ) (42) 1 ⊕ ⊕ m ⊂ ⊂ 0 1 ⊕ ⊕ m n 1 θ K . . . θ K C 1+ c ε 95n (θ K . . . θ K ) (43) 1 1 ⊕ ⊕ m m ⊂ ⊂ 0 1 1 ⊕ ⊕ m m 20 where c0 > 1 is an absolute constant. First we assume that n 19n ε < γ− n− (44) for a suitable absolute constant γ > 1 where γ is a chosen in a way such that 1 1 c˜nnnε 19 < (45) 2n+1 for the constantc ˜ of Corollary 5.4. We have

1 1 1 1 1 1 V (K 2 C 2 ) V K + C (1 + ε)V (K) 2 V (C) 2 ; · ≤ 2 · 0 2 · ≤ therefore, Corollary 5.4 yields a positive deﬁnite diagonal matrix Φ such that

n n 1 1 n n 1 V ((ΦK)∆C) < c˜ n ε 19 V (C) and V (K∆(Φ− C)) < c˜ n ε 19 V (K) (46) wherec> ˜ 1 is an absolute constant. Let 1 M = K (Φ− C), ∩ and hence (46) yields that

1 V (M) > (1 c˜nnnε 19 )V (K) (47) − 1 V (ΦM) > (1 c˜nnnε 19 )V (C). (48) − As M K and ΦM C, it follows that ⊂ ⊂ 1 1 1 1 1 1 1 2 2 n n 19 2 2 V 2 M +0 2 (ΦM) (1 + ε)V (K) V (C) (1 + 2˜c n ε )V (M) V (ΦM) ≤ 1 1 ≤ 1 = (1 + 2˜cnnnε 19 )V (M 2 (ΦM) 2 ). · Now we apply Corollary 5.5, and deduce the existence of an absolute con- − 5 1 1 stant c1 > 0 such that either sΦ In c1n ε 95n for s = (det Φ) n , or k − k∞ ≤ · there exist s ,...,s > 0 and a partition of 1,...,n into proper subsets 1 m { } J ,...,J , m 2, such that 1 m ≥ m 5 1 (L M) 1+ c n ε 95n M Jk ∩ ⊂ 1 · Mk=1 where for k = 1,...,m, we have

5 1 s (L M) Φ(L M) 1+ c n ε 95n s (L M). k · Jk ∩ ⊂ Jk ∩ ⊂ 1 · k · Jk ∩ 21 We deduce from (45), (47), (48), and Lemma 5.3 the existence of an absolute constant c2 > 1 that

1 M K (1 + c2nε 19n )M ⊂ ⊂ 1 ΦM C (1 + c nε 19n )ΦM. ⊂ ⊂ 2 5 1 Now if sΦ In c1n ε 95n , then we can choose m = 1 and K1 = M k − k∞ ≤ · 5 1 to verify Theorem 2.3. On the other and, if sΦ In l > c1n ε 95n , then k − k∞ · we choose

1 5 1 K = 1+ c n ε 95n − (L M) for k = 1,...,m. k 1 · Jk ∩ For c3 = c1 + c2 + c1c2 and c4 = c1 + c3 + c1c3, it follows that

m m 1 1 K M K (1 + c nε 19n )M (1 + c nε 19n ) (L M) k ⊂ ⊂ ⊂ 2 ⊂ 2 Jk ∩ k=1 k=1 M m M 5 1 1+ c n ε 95n K ⊂ 3 · k k=1 m m M 1 s K ΦK ΦM C (1 + c nε 19n )ΦM k k ⊂ k ⊂ ⊂ ⊂ 2 k=1 k=1 M M m 1 . (1 + c nε 19n ) Φ(L M) ⊂ 2 Jk ∩ k=1 Mm 1 5 1 (1 + c nε 19n ) 1+ c n ε 95n s (L M) ⊂ 2 1 · k · Jk ∩ k=1 M m m 5 1 5 1 1+ c n ε 95n s (L M) 1+ c n ε 95n s K . ⊂ 3 · k Jk ∩ ⊂ 4 · k k Mk=1 Mk=1 1 n 19n This proves Theorem 2.3 if λ = 2 and ε < γ− n− (cf. (44)). 1 Still keeping λ = 2 , we observe that if Q is any unconditional convex body in Rn, then n (Re Q) nQ. (49) i ∩ ⊂ Mi=1 Therefore, if ε γ nn 19n (cf. (44)) holds in (41), then (42) and (43) ≥ − − readily hold for suitable absolute constant c0 > 1 by taking m = n, Kk = 1 (Re K), and choosing θ > 0 in a way such that θ (Re K)= Re C n k ∩ k k k ∩ k ∩

22 1 for k = 1,...,n. In particular, Theorem 2.3 has been veriﬁed if λ = 2 .

Next, we assume that λ [τ, 1 τ] holds for some τ (0, 1 ] in Theo- ∈ − ∈ 2 rem 2.3. First let ε τ. Since ≤ ϕ(t)= V ((1 t) K + t C) − · 0 · is log-concave on [0, 1] according to Lemma 5.1, Lemma 5.2 yields that ε ϕ 1 1+ ϕ(0)ϕ(1); 2 ≤ min 1 λ, λ { − } p or in other words,

1 1 ε 1 1 V K + C 1+ V (K) 2 V (C) 2 . 2 · 0 2 · ≤ τ We deduce from (42) and (43) that for m 1, there exist θ ,...,θ > 0 ≥ 1 m and unconditional compact convex sets K1,...,Km > 0 such that lin Ki, i = 1,...,m, are complementary coordinate subspaces, and

1 ε 95n K . . . K K 1+ cn (K . . . K ) (50) 1 ⊕ ⊕ m ⊂ ⊂ 0 τ 1 ⊕ ⊕ m 1 ε 95n θ K . . . θ K C 1+ cn (θ K . . . θ K (51)). 1 1 ⊕ ⊕ m m ⊂ ⊂ 0 τ 1 1 ⊕ ⊕ m m Finally, if λ [τ, 1 τ] holds for some τ (0, 1 ] in Theorem 2.3 and ∈ − ∈ 2 ε τ, then choosing again m = n, K = 1 (Re K), and θ > 0 in a way ≥ k n k ∩ k such that θk(Rek K)= Rek C for k = 1,...,n, (49) yields (50) and (51). ✷ ∩ ∩

6 Convex bodies and simplicial cones

In this section, we consider the part of a convex body in a Weyl chamber.

LEMMA 6.1 Let H ,...,H be independent linear (n 1)-dimensional 1 n − subspaces, and let W be the closure of a connected component of Rn (H \ 1 ∪ . . . H ). ∪ n (i) If M is a convex body in Rn symmetric through H ,...,H , then ν 1 n M,q ∈ W for any q W ∂ M, and in turn ∈ ∩ ′ M W = x W : x, u h (u) u W . ∩ { ∈ h i≤ M ∀ ∈ }

23 (ii) If λ (0, 1) and K and C are convex bodies in Rn symmetric through ∈ H1,...,Hn, then

1 λ λ W ((1 λ)K + λC)= x W : x, u h (u) − h (u) u W . ∩ − 0 { ∈ h i≤ K C ∀ ∈ } Proof: For (i), it is sufficient to prove the first statement; namely, if q ∈ int W ∂ K, then ν W . ∩ ′ M,q ∈ Let u Sn 1, i = 1,...,n, such that W = x Rn : x, u 0 , and i ∈ − { ∈ h ii≥ } hence q, u > 0, i = 1,...,n, and (i) is equivalent with the statement that h ii if i = 1,...,n, then u ,ν 0. (52) h i K,qi≥ Since q = q 2 q, u u is the reflexted image of q through H , we have ′ − h ii i i q M; therefore, ′ ∈

0 ν ,q q′ = ν , 2 q, u u = 2 q, u ν , u . ≤ h K,q − i h K,q h ii ii h ii · h K,q ii As q, u > 0, we conclude (52), and in turn (i). h ii For (ii), let M = (1 λ)K + λC, and let − 0 1 λ λ M = x W : x, u h (u) − h (u) u W . + { ∈ h i≤ K C ∀ ∈ } Readily, W M M . Therefore, (ii) follows if for any q ∂ M intW , ∩ ⊂ + ∈ ′ ∩ we have q ∂M . As q ∂M intW , there exists u Sn 1 such that ∈ + ∈ ∩ ∈ − q, u = h (u)1 λh (u)λ. Since q ∂ M W , we have u = ν , and hence h i K − C ∈ ′ ∩ M,q (i) yields that ν W . Therefore q ∂M , proving Lemma 6.1 (ii). ✷ M,q ∈ ∈ + The main idea in order to use the known results about unconditional Rn convex bodies is to linearly transfer a Weyl chamber W into the corner 0. ≥ LEMMA 6.2 Let K be a convex body in Rn with o int K, let independent n ∈ v1, . . . , vn R satisfy that vi, vj 0 for 1 i j n, let W = ∈ h Rn i ≥ ≤ R≤ ≤ pos v1, . . . , vn , and let ΦW = 0 for a Φ GL (n, ). { } ≥ ∈ t Rn (i) Φ− W 0. ⊂ ≥ (ii) If ν W for any x W ∂ K, then K,x ∈ ∈ ∩ ′ Rn Rn νΦK,z 0 for any z 0 ∂′ΦK; (53) ∈ ≥ ∈ ≥ ∩

(iii) and there exists an unconditional convex body K0 such that Rn 0 K0 = Φ(W K). ≥ ∩ ∩

24 n Proof: Let e1,...,en be the standard orthonormal basis of R indexed in a way such that ei =Φvi. First we claim that

t e , Φ− v 0 for v W and i = 1,...,n. (54) h i i≥ ∈ Since v = n λ v for λ , . . . , λ 0, we deduce from v , v 0 that j=1 j j 1 n ≥ h j ii≥ P n t t 0 λjvj, vi = v, vi = Φ− v, Φvi = Φ− v, ei , ≤ * + h i h i h i Xj=1 proving (54). In turn, we deduce (i) from (54). If z W ∂ K, then ν W and Φ tν is an exterior normal to ∈ ∩ ′ K,z ∈ − K,z ΦK at Φz, therefore, (ii) follows from (i). Rn Now (53) yields that if z = (z1,...,zn) 0 ∂′ΦK and 0 yi zi, ∈ ≥ ∩ ≤ ≤ i = 1,...,n, then y = (y1,...,yn) ΦK. Therefore repeatedly reﬂecting Rn ∈ 0 ΦK through the coordinate hyperplanes, we obtain the unconditional ≥ ∩ Rn Rn ✷ convex body K0 such that 0 K0 = 0 ΦK = Φ(W K). ≥ ∩ ≥ ∩ ∩

7 Some properties of Coxeter groups

Since if a linear map A leaves a convex body K invariant, then the minimal volume Loewner ellipsoid is also invariant under A, Barthe, Fradelizi [11] 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 Rn are invariant under linear reﬂections A1,...,An through n independent linear (n 1)-planes H ,...,H , then there exists B SL(n) such that BA B 1,...,BA B 1 − 1 n ∈ 1 − n − are orthogonal reﬂections through BH1,...,BHn and leave BK and BC invariant.

For the theory of Coxeter groups, we follow Humpreys [59]. 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 reﬂections through p ,...,p for independent p ,...,p V . A linear subspace L 1⊥ n⊥ 1 n ∈ of V is invariant under G if and only if p ,...,p L L . We say that 1 n ∈ ∪ ⊥ an invariant linear subspace L is irreducible if L = o and any invariant 6 { } subspace L L satisﬁes either L = L or L = o , and hence the action ′ ⊂ ′ ′ { } of G on an irreducible invariant subspace is irreducible. Since the intersec- tion and the orthogonal complement of invariant subspaces is invariant, the

25 irreducible subspaces L ,...,L , m 1 are pairwise orthogonal, and 1 m ≥ L . . . L = V. (55) 1 ⊕ ⊕ m It follows that any A G can be written as A = A . . . A . For an ∈ |L1 ⊕ ⊕ |Lm invariant subspace L V , we set G = A : A G , and write O(L) ⊂ |L { |L ∈ } to denote the group of isometries of L ﬁxing the origin. In particular, our main task is to understand irreducible Coxeter groups.

LEMMA 7.2 (Barthe, Fradelizi) Let G be closure of the Coxeter group generated by the orthogonal reflections through p1⊥,...,pn⊥ for independent p ,...,p Rn. If L Rn is an irreducible invariant subspace, and G is 1 n ∈ ⊂ |L infinite, then G = O(L). |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 is finite, then |L a more detailed analysis is needed. To set up the correponding notation, let G = G be the finite Coxeter group generated by some orthogonal ′ |L reflections acting on L. Let H ,...,H L be the linear (d 1)-dimensional 1 k ⊂ − subspaces such that the reflections in G′ are the ones through H1,...,Hk, and let u , . . . , u L o be a system of roots for G ; namely, there are 1 2k ∈ \{ } ′ exactly two roots orthogonal to each Hi, and these two roots are opposite. We note that for algebraic purposes, one usually normalize the roots in a 2 ui,uj way such that h i is an integer but we drop this condition because we 2 ui,ui are only interestedh ini the cones determined by the roots. Let W be the closure of a Weyl chamber; namely, a connected component of L H ,...,H . It is known (see [59]) that \{ 1 k} d W = pos v1, . . . , vd = λivi : λi 0 { } ( ∀ ≥ ) Xi=1 where v , . . . , v L are independent. In addition, for any x L H ,...,H , 1 d ∈ ∈ \{ 1 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 H1,...,Hk and u1, . . . , u2k in a way such that Hi = ui⊥ for i = 1,...,d are the ”walls” of W , and u , v > 0 for i = 1,...,d; h i ii (56) u , v = 0 for 1 i < j d. h i ji ≤ ≤ In this case, reflections L L through H ,...,H generate G , and u , . . . , u → 1 d ′ 1 d is called a simple system of roots. The order we list simple roots is not related to the corresponding Dynkin diagram.

26 LEMMA 7.3 Let G be the Coxeter group generated by the orthogonal reflections through p ,...,p for independent p ,...,p Rn. If L Rn 1⊥ n⊥ 1 n ∈ ⊂ is an irreducible invariant d-dimensional subspace with d 2, and G is ≥ |L finite, and W = pos v , . . . , v L is the closure of a Weyl chamber for { 1 d} ⊂ G L, then | 1 v , v v v . (57) h i ji≥ d · k ik · k jk Proof: Let G = G . We use the classification of finite irreducible Coxeter ′ |L groups. For the cases when G′ is either of Dd, E6, E7, E8 (see Adams [1] about E6, E7, E8), we use the known simple systems of roots in terms of the orthonormalt basis e1,...,ed of L to construct v1, . . . , vd via (56). However, there is a unified construction for the other finite irreducible Coxeter groups because they are the symmetries of some regular polytopes.

Case 1: G′ is one of the types I2(m), Ad, Bd, F4, H3, H4 In this case, G′ is the symmetry group of some d-dimensional regular polytope P centered at the origin. Let F0 . . . Fd 1 be a tower of faces ⊂ ⊂ − of P where dim F = i, i = 0,...,d 1. Deﬁning v to be the centroid of i − i Fi 1, i = 1,...,d, we have that W = pos v1, . . . , vd is the closure of a − { } Weyl chamber because the symmetry group of P is simply transitive on the towers of faces of P . As G′ is irreducible, the John ellipsoid of P (the unique ellipsoid of largest volume contained in P ) is a d-dimensional ball centered at the origin of some radius r > 0. It follows that P drBn, and hence r v dr ⊂ ≤ k ik ≤ for i = 1,...,d. In addition, vi is the closest point of aﬀ Fi 1 to the origin − for i = 1,...,d, and vj Fi 1 if 1 j i, thus vj, vi = vi, vi if ∈ − ≤ ≤ h i h i 1 j i d. We conclude that if 1 j i d, then ≤ ≤ ≤ ≤ ≤ ≤ v , v v 1 h j ii = k ik . v v v ≥ d k jk · k ik k jk

Case 2: G′ = Dn In this case, a simple system of roots is

u = e e for i = 1,...,d 1, i i − i+1 − ud = ed 1 + ed. −

In turn, we may choose v1, . . . , vd as i vi = l=1 el for i = 1,...,d 2 and i = d, d 1 − vd 1 = vd + l=1− el. − −P P 27 As v , v is a positive integer for i = j, and v √d for i = 1,...,d, we h i ji 6 k ik≤ conclude (57).

Case 3: G′ = E6 In this case d = 6, and a simple system of roots is

u = e e for i = 1, 2, 3, 4, i i − i+1 u5 = e4 + e5 u = √3 e 5 e . 6 6 − l=1 l P Using coordinates in e1,...,e6, we may choose v1, . . . , v6 as v1 = (√3, 0, 0, 0, 0, 1), v = (√3, √3, 0, 0, 0, 2), v = (√3, √3, √3, 0, 0, 3), v = (1, 1, 1, 1, 1, √3), 2 3 4 − v = (1, 1, 1, 1, 1, 5 ) and v = (0, 0, 0, 0, 0, 3). As v , v 3 for i = j, and 5 √3 6 h i ji≥ 6 v √18 for i = 1,..., 6, we conclude (57). k ik≤

Case 4: G′ = E7 In this case d = 7, and a simple system of roots is

u = e e for i = 1, 2, 3, 4, 5, i i − i+1 u6 = e5 + e6 u = √2 e 6 e . 7 7 − l=1 l P Using coordinates in e1,...,e7, we may choose v1, . . . , v7 as v1 = (2, 0, 0, 0, 0, 0, √2), v = (1, 1, 0, 0, 0, 0, √2), v = (1, 1, 1, 0, 0, 0, 3 ), v = (1, 1, 1, 1, 0, 0, 2√2), 2 3 √2 4 v = (1, 1, 1, 1, 1, 1, 2√2), v = (1, 1, 1, 1, 1, 1, 3√2) and v = (0, 0, 0, 0, 0, 4). 5 − 6 7 As v , v 4 for i = j, and v < √28 for i = 1,..., 7, we conclude (57). h i ji≥ 6 k ik

Case 5: G′ = E8 In this case d = 8, and a simple system of roots is

u = e e for i = 1, 2, 3, 4, 5, 6, 7, i i − i+1 u = 5 e + 8 e . 8 − l=1 l l=6 l Using coordinates in eP,...,e , weP may choose v , . . . , v as v = (1, 1, 1, 1, 1, 1, 1, 1), 1 8 1 8 1 − − − − − − − v = (0, 0, 1, 1, 1, 1, 1, 1), v = ( 1, 1, 1, 3, 3, 3, 3, 3), 2 − − − − − − 3 − − − − − − − − v = ( 1, 1, 1, 1, 2, 2, 2, 2), v = ( 1, 1, 1, 1, 1, 5 , 5 , 5 ), 4 − − − − − − − − 5 − − − − − − 3 − 3 − 3 v = ( 1, 1, 1, 1, 1, 1, 2, 2), v = ( 1, 1, 1, 1, 1, 1, 1, 3) 6 − − − − − − − − 7 − − − − − − − − and v = ( 1, 1, 1, 1, 1, 1, 1, 1). As v , v 6 for i = j, and 8 − − − − − − − − h i ji ≥ 6 v < √48 for i = 1,..., 8, we conclude (57). ✷ k ik

28 For a convex body invariant under a Coxeter group, we can determine the some exterior normal at certain points provided by the symmetries of the convex body.

LEMMA 7.4 Let G be the closure of a Coxeter group generated by n independent orthogonal reflections of Rn, let L Rn be an irreducible linear ⊂ subspace and let K be a convex body in Rn invariant under G. (i) If G is finite, and W = pos v , . . . , v L is the closure of a Weyl |L { 1 d}⊂ chamber for G , and t v ∂K for t > 0, i = 1, ldots, d, then v is |L i i ∈ i i an exterior normal at tvi. (ii) If G is infinite and v L o , and tv ∂K for t> 0, then v is an |L ∈ \{ } ∈ exterior normal at tv.

Proof: Let d = dim L. For (i), ﬁrst we claim that there exist independent u1, . . . , un 1 vi⊥ such − ∈ that the reﬂection through u lies in G for j = 1,...,n 1. To construct j⊥ − u1, . . . , un 1 vi⊥, if d 2, then we choose roots u1, . . . , ud 1 vi⊥ for G L − ∈ ≥ − ∈ | that corresponds to the walls of W containing vi. In addition, if d

8 The proof Theorem 1.4

Lemma 7.1 and the linear invariance of the L0-sum yield that we may assume that A ,...,A are orthogonal reﬂections through the linear (n 1)-spaces 1 n − H ,...,H , respectively, with H . . . H = o where K and C are 1 n 1 ∩ ∩ n { } invariant under A1,...,An.

29 Let G be the closure of the group generated by A1,...,An, and let n L1,...,Lm be the irreducible invariant subspaces of R of the action of G. If t ,...,t > 0 and Ψ GL(n, R) satisﬁes Ψx = t x for x L and 1 m ∈ i ∈ i i = 1,...,m, then

ΨK and ΨC are both invariant under G. (58)

Let E be the John ellipsoid of K, that is, the unique ellipsoid of maximal volume contained in K. Therefore, E is also invariant under G. In particular, we can choose the principal directions of E in a way such that each is contained in one of the L , and L E is a Euclidean ball of dimension i i ∩ dim Li. Therefore, after applying a suitable linear transformation like in (58), we may assume that E = Bn, and hence

Bn K nBn. (59) ⊂ ⊂ For any i = 1,...,n, let G = G if G is finite, and let G be the i |Li |Li i symmetry group of some dim Li dimensional regular simplex in Li centered at the origin if G is infinite. |Li We consider the finite subgroup G G that is the direct sum of G ,...,G , ⊂ 1 m acting in the natural way G = G for i = 1,...,m. Let 0= p < p < |Li i 0 1 ... 0 and u , v = 0 for l = j. Therefore, u , . . . , u are the h j ji h j li 6 1⊥ n⊥

30 walls of W ; namely, the linear hulls of the facest of the simplicial cone W , and the reﬂections through u1⊥, . . . , un⊥ are symmetries of both K and C (and actually generate G). We may apply Lemma 6.2 to W because of Lemma 6.1, (60) and (61), and deduce the existence unconditional convex bodies K and C such thate Rn Rn 0 K = Φ(W K) and 0 C = Φ(W C). e ≥e ∩ ∩ ≥ ∩ ∩ We claim that e e Rn 0 ((1 λ)K + λC) Φ (W ((1 λ)K +0 λC)) . (63) ≥ ∩ − ⊂ ∩ − t Rn According to Lemma 6.1 and to Φ− W 0 (cf. Lemma 6.2), we have e e ⊂ ≥ Rn Rn 1 λ 1 λ Rn 0 ((1 λ)K + λC) = x 0 : x, u hKe (u) − hCe(u) − u 0 ≥ ∩ − { ∈ ≥ h i≤ ∀ ∈ ≥ } n 1 λ λ t R e e x 0 : x, u hK (u) − hC (u) u Φ− W . e e ⊂ { ∈ ≥ h i≤ ∀ ∈ } t Rn Rn We observe that if u Φ− W , then there exist y0 0 ∂K = 0 ∂(ΦK) n ∈ n ∈ ≥ ∩ ≥ ∩ and z R ∂C = R ∂(ΦC) with h e (u)= y , u and h e(u)= z , u . 0 ∈ 0 ∩ 0 ∩ K h 0 i C h 0 i For v = Φt≥u W , y =≥ Φ 1y W ∂K and y = Φ 1ye W ∂K, it ∈ − 0 ∈ ∩ − 0 ∈ ∩ follows that v is ane exterior normal to K at y and to C at z, and

1 λ λ t 1 λ t λ 1 λ λ 1 λ λ h e (u) − h e(u) = Φy, Φ− v − Φz, Φ− v = y, v − z, v = h (v) − h (v) . K C h i h i h i h i K C We deduce from the considerations just above and from applying Lemma 6.1 to W that Rn 1 λ λ 0 ((1 λ)K + λC) Φ q W : q, v hK (v) − hK (v) v W ≥ ∩ − ⊂ { ∈ h i≤ ∀ ∈ } = Φ (W ((1 λ)K +0 λC)) , e e ∩ − proving (63). Writing G to denote the cardinality of G, (62) yields | | V (M)= G V (M W ) e | | · e∩ where M is either K, C or (1 λ) K + λ C. We deduce from (63) and − e· 0 · the condition in Theorem 1.4 that n Rn V ((1 λ) K +0 λ C) = 2 V 0 ((1 λ) K +0 λ C) − · · ≥ ∩ − · · n 2 V (Φ (W ((1 λ)K +0 λC))) e e ≤ n ∩ − e e 2 detΦ 1 λ λ | | (1 + ε)V (K) − V (C) ≤ G · | | 1 λ λ = (1+ ε)V (K) − V (C) . e 31 e e We apply the following equivalent form of Theorem 2.3 to K and C where λ [τ, 1 τ] for τ (0, 1 ]. There exist absolute constantc ˜ > 1, comple- ∈ − ∈ 2 mentary coordinate linear subspaces Λ ,..., Λ , k 1, withe k Λe = Rn 1 k ≥ ⊕j=1 j such that 1 ε 95n k K Λ e1 +c ˜n e K, e (64) ⊕j=1 ∩ j ⊂ τ and there exist θ1,...,θk >e 0 suche that e

1 ε 95n k θ K Λ C 1 +c ˜n k θ K Λ . (65) ⊕j=1 j ∩ j ⊂ ⊂ τ ⊕j=1 j ∩ j 1 e e e e e For Λj =Φ− Λj, j = 1,...,k, we deduce that

k 1 e ε 95n W (K Λ ) 1 +c ˜n (W K), (66) ∩ ∩ j ⊂ τ ∩ Xj=1 and

k 1 k ε 95n W θ (K Λ ) W C 1 +c ñ W θ (K Λ ) . ∩ j ∩ j ⊂ ∩ ⊂ τ  ∩ j ∩ j  Xj=1 Xj=1  (67) We observe that each Λj is spanned by a subset of v1, . . . , vn. For the rest of the argument, first we assume that ε is small enough to satisfy 1 ε 95n 1 cñ < . (68) τ n2 We claim that if (68) holds, then

each Λj, j = 1,...,k, is invariant under G. (69)

We suppose indirectly that the claim (69) does not hold, and we seek a contradiction. In this case, k 2. Since each Λ is spanned by a subset of ≥ j v1, . . . , vn, after possibly reindexing L1,...,Lm, Λ1,..., Λk and v1, . . . , vn, we may assume that v L Λ and v L Λ . For i = 1,...,n, let 1 ∈ 1 ∩ 1 2 ∈ 1 ∩ 2 s > 0 satisfy s v ∂K; therefore, (59) yields i i i ∈ 1 s n, (70) ≤ i ≤ and hence s v L K Λ and v L K Λ . (71) 1 1 ∈ 1 ∩ ∩ 1 2 ∈ 1 ∩ ∩ 2

32 It follows from (60) that 1 v , v . (72) h 1 2i≥ n We deduce from (71), and then from (66) that

k 1 ε 95n s v + v W (K Λ ) 1 +c ˜n (W K). (73) 1 1 2 ∈ ∩ ∩ j ⊂ τ ∩ Xj=1

Lemma 7.4 yields that v1 is an exterior unit normal to ∂K at s1v1, and hence s1 = hK (v1). We deduce from ﬁrst (73) and then from assumption (68) and the formula (70) that

1 ε 95n s + v , v = v ,s v + v 1 +c ˜n h (v ) 1 h 1 2i h 1 1 1 2i≤ τ K 1 1 ε 95n 1 = s +c ˜n s < 1+ . (74) 1 τ 1 n On the other hand, we have s + v , v 1+ 1 by (72), contradicting (74). 1 h 1 2i≥ n In turn, we conclude (69) under the assumption (68). We deduce from (66), (67), (69) and the symmetries of K and C that

1 ε 95n k (K Λ ) 1 +c ˜n K, (75) ⊕j=1 ∩ j ⊂ τ and

1 ε 95n k θ (K Λ ) C 1 +c ˜n k θ (K Λ ) . (76) ⊕j=1 j ∩ j ⊂ ⊂ τ ⊕j=1 j ∩ j In addition, the symmetries of K and (69) yield that K Λ = K Λ for ∩ j | j j = 1,...,k, therefore, K k (K Λ ) . ⊂⊕j=1 ∩ j Combining this relation with (75) and (76) implies Theorem 1.4 under the assumption (68). Finally, we assume that

1 ε 95n 1 c˜n , (77) τ ≥ n2 and hence 1 ε 95n (4˜c)n n2. (78) τ ≥ 33 For i = 1,...,m, the symmetries of K and C yield that r (Bn L ) is i ∩ i the John ellipsoid of K L and θ r (Bn L ) is the John ellipsoid of C L ∩ i i i ∩ i ∩ i for some r ,θ > 0. For K = ri (Bn L ), i = 1,...,m, we have i i i n ∩ i m K conv mK ,...,mK ; ⊕i=1 i ⊂ { 1 m} therefore, it follows from (78) that

1 ε 95n m K K n2 m K 1 + (4˜c)n m K ⊕i=1 i ⊂ ⊂ ·⊕i=1 i ⊂ τ ⊕i=1 i 1 ε 95n m θ K C n2 m θ K 1 + (4˜c)n m θ K , ⊕i=1 i i ⊂ ⊂ ·⊕i=1 i i ⊂ τ ⊕i=1 i i proving Theorem 1.4 under the assumption (77). ✷

9 Proof of Theorem 1.5

As in the case of Theorem 1.4, it follows from Lemma 7.1 and the linear invariance of the L0-sum that we may assume that A1,...,An are orthogonal reﬂections through the linear (n 1)-spaces H ,...,H , respectively, with − 1 n H . . . H = o where K and C are invariant under A ,...,A . We 1 ∩ ∩ n { } 1 n write G to denote the closure of the group generated by A1,...,An, and n L1,...,Lm to denote the irreducible invariant subspaces of R of the action of 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, we may assume that V (K)= V (C) = 1. In this case, the condition in Theorem 1.5 states that

hC log dVK < ε (79) n−1 h ZS K for ε> 0. First we assume that nε < 1, (80) for t [0, 1], we deﬁne ∈ ϕ(t)= V ((1 t) K + t C). − · 0 ·

34 According to (3.7) in B¨or¨oczky, Lutwak, Yang, Zhang [20], we have

hC ϕ′(0) = n log dVK, (81) n−1 h ZS K and hence (79) and the assumption (80) yield that ϕ′(0) < nε where nε < 1. We deduce from Lemma 5.2 (ii) that

1 1 1 V K + C = ϕ < 1+ nε. 2 · 0 2 · 2 Now we apply Theorem 1.4, and conclude that for some m 1, there exist ≥ θ1,...,θm > 0 and compact convex sets K1,...,Km > 0 invariant under G such that lin Ki, i = 1,...,m, are complementary coordinate subspaces, and

n 1 K . . . K K 1+ c ε 95n (K . . . K ) (82) 1 ⊕ ⊕ m ⊂ ⊂ 1 ⊕ ⊕ m n 1 θ K . . . θ K C 1+ c ε 95n (θ K . . . θ K ) (83) 1 1 ⊕ ⊕ m m ⊂ ⊂ 1 1 ⊕ ⊕ m m where c> 1 is an absolute constant. In turn, we deduce Theorem 1.5 under the assumption nε < 1 on (80). On the other hand, if 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 Richard Gardner for illuminating discussions.

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