The ratio between the volume of the p-centroid body of a convex body K in R n and the volume of K attains its minimum value if and only if K is an origin symmetric ellipsoid. This result, the L p -Busemann-Petty centroid inequality, was recently proved by Lutwak, Yang, and Zhang. In this paper we show that all the intrinsic volumes of the p-centroid body of K are convex functions of a time-like parameter when K is moved by shifting all the chords parallel to a fixed direction. The L p -Busemann-Petty centroid inequality is a consequence of this general fact.

Abstract. 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.

Abstract. Basic properties of finite subsets of the integer lattice Z n 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.

Abstract. There are sequences of directions such that, given any compact set K ⊂ R n , the sequence of iterated Steiner symmetrals of K in these directions 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 directions 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.

Abstract. 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-Santaló inequality for two-dimensional convex sets.

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.

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