Julio Bernués

An extension of Milman's reverse Brunn-Minkowski inequality

compact convex sets, followed by an analytical proof by Minkowski [Min]. The inequality (1)for compact sets, not necessarily convex, was first proved by Lusternik [Lu]. A very simple proof ofit can be found in [Pi 1], Ch. 1.It is easy to see that one cannot expect the reverse inequality to hold at all, even if it is perturbedby a fixed constant and we restrict ourselves to balls (i.e. convex symmetric compact sets with theorigin as an interior point). Take for instance A

THE THEOREMS OF CARATHEODORY AND GLUSKIN FOR 0 < p < 1

In this note we prove the p-convex analogue of both Caratheodorys convexity theorem and Gluskins theorem concerning the diameter of Minkow- ski compactum. Throughout this note X will denote a real vector space and p will be a real number, 0 0, with Xp + pp = 1. Given A c X, the p-convex hull of A is defined as the intersection of all p-convex sets that contain A. This set is denoted by p-conv(A). A (real) p-normed space (X, \\ • ||) is a (real) vector space equipped with a quasi-norm such that ||x + v||p en for some absolute constant c. Our purpose is to study this problem in the p-convex setting. In (Pe), Peck gave an upper bound of the diameter of JAP , namely, diam(^f ) < n2/p~x. We will show that this bound is optimal (Theorem 2). When proving it, in order to compute some volumetric estimates, it will be necessary to have the corresponding version for p < 1 of Caratheodorys convexity theorem (Theorem 1). The results of this note are the following:

Factoring Sobolev inequalities through classes of functions

We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, [21], [3], while the second one relies on tools from Convex Geometry, [32], [16]. In this paper we prove a (sharp) connection between them.