Paul Goodey

Zonoids and Generalisations

Publisher Summary This chapter focuses on zonoids and generalizations and discusses analytic aspects of centrally symmetric convex bodies. Zonoids are limits of zonotopes in the Hausdorff metric. Zonoids occur as images of a natural operation—namely, as projection bodies. Zonoids easily allow extending some analytic notions, properties, and formulae to more general convex sets (generalized zonoids) or even to all centrally symmetric convex bodies. The characterization of surface area measures of convex bodies shows that centered zonoids are precisely the projection bodies of convex bodies. A centrally symmetric polytope with nonsymmetric faces, for example, an octahedron, is not a zonoid. The only polytopal generalized zonoids are the zonotopes. The generalized zonoids are dense in the centrally symmetric bodies. Any sufficiently smooth centrally symmetric body should be a generalized zonoid. There is a hierarchy of centrally symmetric convex bodies corresponding to the nature of the generating distribution. Zonotopes are the bodies whose generating distribution is an atomic measure. Zonoids are those for which it is a positive measure, and the generalized zonoids correspond to the case of signed measures.

Intersection bodies and ellipsoids

In this paper we study various classes of centrally symmetric sets in d -dimensional Euclidean space R d . As we will see, it is appropriate to focus our attention on those sets which have interior points.

Processes of flats induced by higher dimensional processes

Stationary processes of k-flats in \(\mathbb{E}\) d can be thought of as point processes on the Grassmannian \(\mathcal{L}\) k d of k-dimensional subspaces of \(\mathbb{E}\) d . If such a process is sampled by a (d−k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d−1 and j=0,...,2[r/2]−1, where r is the rank of the manifold \(\mathcal{L}\) k d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.

The determination of convex bodies from the mean of random sections

Random sectioning of particles (compact sets in ℝ 3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [ 23 ] or the book [ 20 ]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection X k = K ∩ E is a ( k -dimensional, k = 1 or 2) random set. It is clear that the distribution of X k determines K uniquely and that if E 1 ,…, E n are such flats, the most natural estimator for K would be the convex hull

Translative integral formulae for convex bodies.

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.

Section and projection means of convex bodies

This paper is concerned with various geometric averages of sections or projections of convex bodies. In particular, we consider Minkowski and Blaschke sums of sections as well as Minkowski sums of projections. The main result is a Crofton-type formula for Blaschke sums of sections. This is used to establish connections between the different averages mentioned above. As a consequence, we obtain results which show that, in some circumstances, a convex body is determined by the averages of its sections or projections.

Centrally symmetric convex bodies and Radon transforms on higher order Grassmannians

The purpose of this work is to investigate the relationship between Radon transforms and centrally symmetric convex bodies. Because of the injectivity properties of the Radon transform it is natural to consider transforms on the sphere separately from those on the higher order Grassmannians. Here we shall concentrate on the latter, whilst the former will be the subject of another article presently in preparation, Goodey and Weil [1991].

Projection Functions on Higher Rank Grassmannians

This work comprises a study of the behaviour of projection functions of convex bodies considered as functions defined on Grassmannian manifolds. We show that the behaviour in the case of higher rank manifolds is often very different from the rank 1 case. We also study the images of projection functions under Radon transforms. If X is an n-dimensional normed space, and d denotes the Banach-Mazur distance, then d(X, l ∞ n ) ≤ cn 5/6.

RADON TRANSFORMS OF PROJECTION FUNCTIONS

We show that a centrally symmetric convex body is determined, up to translation, by any Radon transform of one of its projection functions. It is also proved that all Radon transforms are injective when restricted to projection functions of polytopes. Both results were known previously in the case of Radon transforms R i,j with i j . Here we use harmonic analysis on Grassmann manifolds to show that these results extend to all Radon transforms.

Integral geometric formulae for projection functions

Principal kinematic and Crofton formulae are established for projection functions of convex bodies. In the case of centrally symmetric bodies, the results are expressed in terms of Radon transforms of the projection functions.