for any compact sets K, T ⊂ R, where (K +T )/2 = {(x+ y)/2; x ∈ K, y ∈ T} is half of the Minkowski sum of K and T , and where V oln stands for the Lebesgue measure in R. Equality in (1) holds if and only if K is a translate of T and both are convex, up to a set of measure zero. The literature contains various stability estimates for the Brunn-Minkowski inequality, which imply that when there is...

In this paper we consider the following analog of Bezout inequality for mixed volumes: V (P1, . . . , Pr,∆ )Vn(∆) r−1 ≤ r ∏ i=1 V (Pi,∆ ) for 2 ≤ r ≤ n. We show that the above inequality is true when ∆ is an n -dimensional simplex and P1, . . . , Pr are convex bodies in R . We conjecture that if the above inequality is true for all convex bodies P1, . . . , Pr , then ∆ must be an n -dimensional...

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

Correspondence On the Similarity of the Entropy Power Inequality The preceeding equations allow the entropy power inequality and the Brunn-Minkowski Inequality to be rewritten in the equivalent form (4) where X' and Y' are independent normal variables with corresponding entropies H(X') = H(X) and H(Y') = H(Y). Verification of this restatement follows from the use of (1) to show that Abstract-Th...

The Brunn-Minkowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This inequality plays a crucial role in the theory of convex bodies and has many interactions with isoperimetry and functional analysis. Stability of optimizers of this inequality in one dimension is a consequence of classical results in additive combinatorics....

We investigate the effect of a Steiner type symmetrization on the isotropic constant of a convex body. We reduce the problem of bounding the isotropic constant of an arbitrary convex body, to the problem of bounding the isotropic constant of a finite volume ratio body. We also add two observations concerning the slicing problem. The first is the equivalence of the problem to a reverse Brunn-Min...

The aim of the present paper is to provide a unitary exposition on possible representations for positively homogeneous functions in the dual space. Starting from the classical Minkowski–Hörmander duality result of convex analysis, we develop a scheme in order to associate to a positively homogeneous function families of sublinear functionals representing it. Such dual description allows us to c...

In this paper, the mixed Lp-surface area measures are defined and Lp Minkowski inequality is obtained consequently. Furthermore, projection for bodies established.

in this paper we study the impact of minkowski metric matrix on a projection in theminkowski space m along with their basic algebraic and geometric properties.the relationbetween the m-projections and the minkowski inverse of a matrix a in the minkowski spacemis derived. in the remaining portion commutativity of minkowski inverse in minkowski spacem is analyzed in terms of m-projections as an a...