Richard J. Gardner

The Brunn-Minkowski inequality

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

A positive answer to the Busemann-Petty problem in three dimensions

We prove that in E3 the Busemann-Petty problem, concerning cen- tral sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (ni 1)dimensional X-ray) gives the ((ni 1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R n and leads to a unifled analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies inR n such that the ((ni 1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive deflnite distributions, our formula shows that the answer to the problem depends on the behavior of the (ni 2)-nd derivative of the parallel section functions. The a‐rmative answer to the Busemann-Petty problem for n• 4 and the negative answer for n‚ 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

Discrete tomography: Determination of finite sets by X-rays

We study the determination of finite subsets of the integer lattice En, n > 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of En (i.e., finite subsets F with F = En n conv F) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammers X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in 22 have the property that convex subsets of 22 are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.

Intersection bodies and the Busemann-Petty problem

It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in ¿-dimensional Euclidean space Ed is negative for a given d if and only if certain centrally symmetric convex bodies exist in Ed which are not intersection bodies. It is also shown that a cylinder in Ed is an intersection body if and only if d < 4 , and that suitably smooth axis-convex bodies of revolution are intersection bodies when d < 4. These results show that the Busemann-Petty problem has a negative answer for d > 5 and a positive answer for d = 3 and d = 4 when the body with smaller sections is a body of revolution.

On the computational complexity of reconstructing lattice sets from their x-rays

Abstract We study the computational complexity of various inverse problems in discrete tomography. These questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. We completely settle the complexity status of the basic problems of existence (data consistency), uniqueness (determination), and reconstruction of finite subsets of the d-dimensional integer lattice γd that are only accessible via their line sums (discrete X-rays) in some prescribed finite set of lattice directions. Roughly speaking, it turns out that for all d ⩾ 2 and for a prescribed but arbitrary set of m ⩾ 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are NP-complete (or NP-equivalent) otherwise.

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.

Uniqueness and Complexity in Discrete Tomography

We study the discrete inverse problem of reconstructing finite subsets of the n-dimensional integer lattice ℤn that are only accessible via their line sums (discrete X-rays) in a finite set of lattice directions. Special emphasis is placed on the question of when such sets are uniquely determined by the data and on the difficulty of the related algorithmic problems. Such questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high-resolution transmission electron microscopy.

Convergence of algorithms for reconstructing convex bodies and directional measures

We investigate algorithms for reconstructing a convex body K in R n from noisy measurements of its support function or its brightness function in k directions u 1 ,..., u k . The key idea of these algorithms is to construct a convex polytope P k whose support function (or brightness function) best approximates the given measurements in the directions u 1 ,...,u k (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian. It is shown that this procedure is (strongly) consistent, meaning that, almost surely, P k tends to K in the Hausdorff metric as k → ∞. Here some mild assumptions on the sequence (u i ) of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the L 2 metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball. Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u 1 ,...,u k . Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.

The dual Orlicz–Brunn–Minkowski theory ☆

This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz- Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is derived and a corresponding dual Orlicz-Minkowski inequality proved. The inequalities proved yield as special cases the precise duals of the conjectured log-Brunn-Minkowski and log-Minkowski inequalities of Boroczky, Lutwak, Yang, and Zhang. A new addition of star sets called radial M-addition is also introduced and shown to relate to the radial Orlicz addition.