Jesús Bastero

Positions of convex bodies associated to extremal problems and isotropic measures

Abstract We show that there are close relations between extremal problems in dual Brunn–Minkowski theory and isotropic-type properties for some Borel measures on the sphere. The methods we use allow us to obtain similar results in the context of Firey–Brunn–Minkowski theory. We also study reverse inequalities for dual mixed volumes which are related with classical positions, such as l-position or isotropic position.

John's Decomposition of the Identity in the Non-Convex Case

We prove an extension of the classical Johns Theorem, that characterises the ellipsoid of maximal volume position inside a convex body by the existence of some kind of decomposition of the identity, obtaining some results for maximal volume position of a compact and connected set inside a convex set with nonempty interior. By using those results we give some estimates for the outer volume ratio of bodies not necessarily convex.

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

Commutators for the maximal and sharp functions

The class of functions for which the commutator with the HardyLittlewood maximal function or the maximal sharp function are bounded on Lq are characterized and proved to be the same. For the Hilbert transform H , and other classical singular integral operators, a well known and important result due to Coifman, Rochberg and Weiss (cf. [2]) states that a locally integrable function b in R is in BMO if and only if the commutator [H, b], defined by [H, b]f = H(bf)− bH(f), is bounded in L, for some (and for all) q ∈ (1,∞). The cancellation implied by the commutator operation and the properties of singular integrals are crucial for the validity of the result. Later in [3], using real interpolation techniques, Milman and Schonbeck proved a commutator result that applies to the Hardy-Littlewood maximal operatorM as well as the sharp maximal operator. In fact the commutator result is valid for a large class of nonlinear operators which we now describe. Let us say that T is a positive quasilinear operator if it is defined on a suitable class of locally integrable functions D(T ) and satisfies i) Tf ≥ 0, for f ∈ D(T ), ii) T (αf) = |α|Tf , for α ∈ R and f ∈ D(T ), iii) |Tf − Tg| ≤ T (f − g), for f, g ∈ D(T ). We have (cf. [3]) Proposition 1. Let b be a nonnegative BMO function and suppose that T is a positive quasilinear operator which is bounded on L(w), for some 1 ≤ q < ∞ and for all w weights belonging to the Muckenhoupt class Ar for some r ∈ [1,+∞). Then [T, b] is bounded on L. In particular the result applies to the maximal operator and the sharp maximal function. Note that since the Hardy-Littlewood maximal operator M is a positive Received by the editors January 6, 1999. 2000 Mathematics Subject Classification. Primary 42B25, 46E30.

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

For the classical Hardy-Littlewood maximal function Mf , a well known and important estimate due to Herz and Stein, gives the equivalence (Mf)∗(t) ∼ f∗∗(t). In the present note, we study the validity of analogous estimates for maximal operators of the form Mp,qf(x) = sup x∈Q ‖fχQ‖p,q ‖χQ‖p,q , where ‖.‖p,q denotes the Lorentz space L(p, q)-norm.

Inequalities for the Gamma function and estimates for the volume of sections of B~p^n

Let B n p = {(x i ) E R n ; Σ n 1 |x i | p < 1} and let E be a k-dimensional subspace of R n . We prove that |E ∩ B n p | 1/k k ≥ |B n p | 1/n n , for 1 ≤ k ≤ (n - 1)/2 and k = n - 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

Elementary reverse Hölder type inequalities with application to operator interpolation theory

We give a very elementary proof of the reverse H6lder type inequality for the classes of weights which characterize the boundedness on LP of the Hardy operator for nonincreasing functions. The same technique is applied to Calderon operator involved in the theory of interpolation for general Lorentz spaces. This allows us to obtain further consequences for intermediate interpolation spaces. 0. INTRODUCTION Arifio and Muckenhoupt characterized the class of weights, w, such that the Hardy operator is bounded on LP(w) for nonnegative and nonincreasing functions (see [AM]). This class, say (AM)p, is composed of those weights for which there is a constant C > 0 such that for every t > 0

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.

A characterization of theMM*-position of a convex body in terms of covariance matrices

We characterize the position of a convex bodyK such that it minimizesM(TK)M*(TK) (theMM*-position) in terms of properties of the measures ‖ · ‖Kdσ(·) and ‖ · ‖K°dσ(·), answering a question posed by A. Giannopoulos and V. Milman. The techniques used allow us to study other extremal problems in the context of dual Brunn-Minkowski theory.