Paolo Gronchi

On the reverse L^p-Busemann-Petty centroid inequality

The volume of the L p -centroid body of a convex body K ⊂ ℝ d is a convex function of a time-like parameter when each chord of K parallel to a fixed direction moves with constant speed. This fact is used to study extrema of some affine invariant functionals involving the volume of the L p -centroid body and related to classical open problems like the slicing problem. Some variants of the L p -Busemann-Petty centroid inequality are established. The reverse form of these inequalities is proved in the two-dimensional case.

A Brunn-Minkowski inequality for the integer lattice

A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.

On volume product inequalities for convex sets

The volume of the polar body of a symmetric convex set K of R d is investigated. It is shown that its reciprocal is a convex function of the time t along movements, in which every point of K moves with constant speed parallel to a fixed direction. This result is applied to find reverse forms of the L p -Blaschke-Santalo inequality for two-dimensional convex sets.

Volume Inequalities for L p -Zonotopes

The classical Minkowski sum of convex sets is defined by the sum of the corresponding support functions. The L p -extension of such a definition makes use of the sum of the p th power of the support functions. An L p -zonotope Z p is the p -sum of finitely many segments and is isometric to the unit ball of a subspace of l q , where 1/ p + 1/ q = 1. In this paper, a sharp upper estimate is given of the volume of Z p in terms of the volume of Z 1 , as well as a sharp lower estimate of the volume of the polar of Z p in terms of the same quantity. In particular, for p = 1, the latter result provides a new approach to Reisners inequality for the Mahler conjecture in the class of zonoids.

Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

AbstractBasic properties of finite subsets of the integer lattice ℤn are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer’s inequality on sections of convex bodies by coordinate hyperplanes.

Extremal convex sets for Sylvester–Busemann type functionals

The Sylvester (d+2)-points problem deals with the probability S(K) that d + 2 random points taken from a convex compact subset K of are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K), the problem of the maximum is still open, unless d = 2. In this article we study generalizations of S(K), which include the Busemann functional – appearing in the formula for the volume of a convex set in terms of the areas of its central sections – and a functional introduced by Bourgain, Meyer, Milman and Pajor in connection with the local theory of Banach spaces. We also show that for these more general functionals ellipsoids are the only minimizers and, in the two-dimensional case, triangles (or parallelograms, in the symmetric case) are maximizers.

Steiner symmetrals and their distance from a ball

It is known that given any convex bodyK ⊂ ℝn there is a sequence of suitable iterated Steiner symmetrizations ofK that converges, in the Hausdorff metric, to a ball of the same volume. Hadwiger and, more recently, Bourgain, Lindenstrauss and Milman have given estimates from above of the numberN of symmetrizations necessary to transformK into a body whose distance from the equivalent ball is less than an arbitrary positive constant.In this paper we will exhibit some examples of convex bodies which are “hard to make spherical”. For instance, for any choice of positive integersn≥2 andm, we construct ann-dimensional convex body with the property that any sequence ofm symmetrizations does not decrease its distance from the ball. A consequence of these constructions are some lower bounds on the numberN.

Symmetrization in geometry

Abstract The concept of an i-symmetrization is introduced, which provides a convenient framework for most of the familiar symmetrization processes on convex sets. Various properties of i-symmetrizations are introduced and the relations between them investigated. New expressions are provided for the Steiner and Minkowski symmetrals of a compact convex set which exhibit a dual relationship between them. Characterizations of Steiner, Minkowski and central symmetrization, in terms of natural properties that they enjoy, are given and examples are provided to show that none of the assumptions made can be dropped or significantly weakened. Other familiar symmetrizations, such as Schwarz symmetrization, are discussed and several new ones introduced.

Convex bodies with extremal volumes having prescribed brightness in finitely many directions

We consider the class of convex bodies in ℝn with prescribed projection (n − 1)-volumes along finitely many fixed directions. We prove that in such a class there exists a unique body (up to translation) with maximumn-volume. The maximizer is a centrally symmetric polytope and the normal vectors to its facets depend only on the assigned directions.Conditions for the existence of bodies with minimumn-volume in the class defined above are given. Each minimizer is a polytope, and an upper bound for the number of its facets is established.

CONVERGENCE IN SHAPE OF STEINER SYMMETRIZATIONS

There are sequences of directions such that, given any compact set KR n , the sequence of iterated Steiner symmetrals of K in these direc- tions converges to a ball. However examples show that Steiner symmetrization along a sequence of directions whose differences are square summable does not generally converge. (Note that this may happen even with sequences of direc- tions which are dense in S n 1 .) Here we show that such sequences converge in shape. The limit need not be an ellipsoid or even a convex set. We also deal with uniformly distributed sequences of directions, and with a recent result of Klain on Steiner symmetrization along sequences chosen from a finite set of directions.