Abstract:
A quantitative version of Minkowski sum, extending the definition of $\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.

Abstract:
A generalization of Young's inequality for convolution with sharp constant is conjectured for scenarios where more than two functions are being convolved, and it is proven for certain parameter ranges. The conjecture would provide a unified proof of recent entropy power inequalities of Barron and Madiman, as well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It is shown that the generalized Brunn-Minkowski conjecture is true for convex sets; an application of this to the law of large numbers for random sets is described.

Abstract:
We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincar\'e type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space.

Abstract:
We establish some analogues of the Brunn-Minkowski inequalities on convex bodies and the Minkowski inequality and their inverse versions. As an application, we generalize and improve some interrelated results.

Abstract:
By using Minkowski addition of convex functions, we prove convexity and rearrangement properties of solutions to some Hessian equations in $\R^3$ and Brunn-Minkowski and isoperimetric inequalities for related functionals.

Abstract:
In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines.

Abstract:
We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Koll\'ar.

Abstract:
We establish some inequalities for the dual -centroid bodies which are the dual forms of the results by Lutwak, Yang, and Zhang. Further, we establish a Brunn-Minkowski-type inequality for the polar of dual -centroid bodies.

Abstract:
The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-Brunn-Minkowski theory, analogous to the way that Orlicz spaces generalize $L_p$ spaces. For appropriate convex functions $\varphi:[0,\infty)^m\to [0,\infty)$, a new way of combining arbitrary sets in $\R^n$ is introduced. This operation, called Orlicz addition and denoted by $+_{\varphi}$, has several desirable properties, but is not associative unless it reduces to $L_p$ addition. A general framework is introduced for the Orlicz-Brunn-Minkowski theory that includes both the new addition and previously introduced concepts, and makes clear for the first time the relation to Orlicz spaces and norms. It is also shown that Orlicz addition is intimately related to a natural and fundamental generalization of Minkowski addition called $M$-addition. The results obtained show, roughly speaking, that the Orlicz-Brunn-Minkowski theory is the most general possible based on an addition that retains all the basic geometrical properties enjoyed by the $L_p$-Brunn-Minkowski theory. Inequalities of the Brunn-Minkowski type are obtained, both for $M$-addition and Orlicz addition. The new Orlicz-Brunn-Minkowski inequality implies the $L_p$-Brunn-Minkowski inequality. New Orlicz-Minkowski inequalities are obtained that generalize the $L_p$-Minkowski inequality. One of these has connections with the conjectured log-Brunn-Minkowski inequality of Lutwak, Yang, and Zhang, and in fact these two inequalities together are shown to split the classical Brunn-Minkowski inequality.

Abstract:
We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of $M$-ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Pl\"unnecke-Ruzsa inequalities from additive combinatorics.