Böröczky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality together with its equality cases for pairs of convex bodies that are both unconditional with respect to some orthonormal basis. Applications of this fact are discussed. Moreover, we prove that the log-Brunn-Minkowski inequality is equivalent to the (B)-Theorem for the uniform measure of the cube (this has been proven by Cordero-Erasquin, Fradelizi and Maurey for the gaussian measure instead).

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesgue measure in dimension n would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension n. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension n, there is a density f n , which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density f n . As byproduct of our methods, we study possible log-concavity of the function t → |(K + p ·e t L) • |, where p ≥ 1 and K, L are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.Conjecture 1.1. (The logarithmic-Brunn-Minkowski inequality) Let K, L be symmetric convex bodies in R n and λ ∈ (0, 1). Then,with equality in the following case: Whenever K = K 1 ×· · ·×K m , for some convex sets K 1 , . . . , K m , that cannot be written as cartesian products of lower dimensional sets, then there exist positive numbers c 1 , . . . , c m , such that LThe conjecture can easily be seen to be wrong for general convex bodies, even for n = 1. Note, also, that for 0The cone-volume measure of K is defined as: S 0 (K, ·) = h K S(K, ·), where S(K, ·) is the surface area measure of K, viewed as a measure on the sphere (see e.g. [41]). In [5], a necessary and sufficient condition was discovered (the planar case was treated by Stancu [42] [43]; see also [6] for other applications of the cone-volume measure and [8] for a possible functional generalization of the classical Minkowski problem). A confirmation of Conjecture 1.1 would answer the following open problem: When do two symmetric convex bodies K, L have proportional cone-volume measures? If Conjecture 1.1 was proven to be true, the pairs (K, L) would be exactly the ones for which equality holds in the log-Brunn-Minkowski inequality. The planar case was settled in [4]: Theorem A. [4] Conjecture 1.1 is true in dimension two.It was shown in [40] that Conjecture 1.1 holds true for pairs of unconditional bodies with respect to the same orthonormal basis. Actually, the proof (based on a result from [9]) shows that this result (as for the inequality) remains true if we replace the Lebesgue measure with any unconditional log-concave measure in R n . Recall that a measure µ is called log-concave if for all convex bodies K, L, it satisfies the Brunn-Minkowski inequality:By a result of C. Borell [3], the absolutely continuous log-concave measures in R n are exactly the ones having log-concave densities, i.e. their logarithms are concave functions. It is reasonable to conjecture the following:Conjecture 1.2. Let µ be an even log-concave measure in R n , K, L be symmetric convex bodies and λ ∈ (0, 1). Then,Conjecture 1.2 is closely connected (actually implies; see Corollary 3....

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence {Π m K} of convex bodies converges to the ball with respect to the Banach-Mazur distance, as m → ∞. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.

Abstract. We consider the following Bezout inequality for mixed volumes:It was shown previously that the inequality is true for any n -dimensional simplex ∆ and any convex bodies K1, . . . , Kr in R n . It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies K1, . . . , Kr in R n . In this paper we prove that this is indeed the case if we assume that ∆ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex n -polytopes. In addition, we show that if a body ∆ satisfies the Bezout inequality for all bodies K1, . . . , Kr then the boundary of ∆ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

In [G. Bianchi, R. J. Gardner and P. Gronchi, Symmetrization in geometry, Adv. Math. 306 2017, 51–88], a systematic study of symmetrization operators on convex sets and their properties is conducted. In the end of their article, the authors pose several open questions. The primary goal of this manuscript is to study these questions.