Stability of the Logarithmic Brunn-Minkowski inequality in the case of many hyperplane symmetries K´aroly J. B¨or¨oczky,∗† Apratim De‡ March 11, 2021 Dedicated to Prof. Erwin Lutwak on the occasion of his seventy-fifth birthday Abstract In the case of symmetries with respect to n independent linear hyperplanes, a stability versions of the Logarithmic Brunn-Minkowski inequality and the Logarithmic Minkowski inequality for convex bodies are established. MSC 52A40 1 Introduction The classical Brunn-Minkowski inequality form the core of various areas in probability, additive combinatorics and convex geometry (see Gardner [48], Schneider [83] and Tao, Vu [85]). For recent related work in the theory of arXiv:2101.02549v3 [math.MG] 10 Mar 2021 valuations, algorithmic theory and the Gaussian setting, see say Jochemko, Sanyal [60, 61], Kane [62], Gardner, Zvavitch [49], Eskenazis, Moschidis [37]. The rapidly developing new Lp-Brunn-Minkowski theory (where p = 1 is the classical case) initiated by Lutwak [68, 69, 70], has become main ∗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] 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 di- rect approach by Esposito, Fusco, Trombetti [38]; therefore, the exponent depended significantly on n. We note that recently, various breakthrough stability results about ge- ometric 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 in- equalities 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, Gu´edon, 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, Sch¨urman, 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, break- through 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 B¨or¨oczky, 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.