Convex Bodies In Space Research Articles

Sumset Estimates in Convex Geometry

Abstract Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic forms of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and the Lebesgue measure on ${\mathbb{R}}^{n}$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the Plünnecke-Ruzsa inequalities. In the last section of the paper, we provide connections of these inequalities to the classical Rogers-Shephard inequality.

On the cells in a stationary Poisson hyperplane mosaic

Abstract Let X be the mosaic generated by a stationary Poisson hyperplane process X̂ in ℝ d . Under some mild conditions on the spherical directional distribution of X̂ (which are satisfied if the process is isotropic), we show that with probability one the set of cells (d-polytopes) of X has the following properties. The translates of the cells are dense in the space of convex bodies. Every combinatorial type of simple d-polytopes is realized infinitely often by the cells of X. A further result concerns the distribution of the typical cell.

Minkowski valuations under volume constraints

We provide a description of the space of continuous and translation invariant Minkowski valuations Φ:Kn→Kn for which there is an upper and a lower bound for the volume of Φ(K) in terms of the volume of the convex body K itself. Although no invariance with respect to a group acting on the space of convex bodies is imposed, we prove that only two types of operators appear: a family of operators having only cylinders over (n−1)-dimensional convex bodies as images, and a second family consisting essentially of 1-homogeneous operators. Using this description, we give improvements of some known characterization results for the difference body.

On the covering index of convex bodies

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize the covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index.

Hölder continuity for support measures of convex bodies

The support measures of a convex body are a common generalization of the curvature measures and the area measures. With respect to the Hausdorff metric on the space of convex bodies, they are weakly continuous. We provide a quantitative improvement of this result, by establishing a H\"older estimate for the support measures in terms of the bounded Lipschitz metric, which metrizes the weak convergence. Specializing the result to area measures yields a reverse counterpart to earlier stability estimates, concerning Minkowski's existence theorem for convex bodies with given area measure.

Local Tensor Valuations

The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant Minkowski tensors that were introduced by McMullen and characterized by Alesker. In analogy to the characterization theorems of Hadwiger and Alesker, we give here a complete classification of all locally defined tensor measures on convex bodies that share with the local Minkowski tensors the basic geometric properties of isometry covariance and weak continuity.

Concentration of random polytopes around the expected convex hull

We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.

Normally Distributed Probability Measure on the Metric Space of Norms

In this paper we propose a method to construct probability measures on the space of convex bodies. For this purpose, first, we introduce the notion of thinness of a body. Then we show the existence of a measure with the property that its pushforward by the thinness function is a probability measure of truncated normal distribution. Finally, we improve this method to find a measure satisfying some important properties in geometric measure theory.

Local tensor valuations on convex polytopes

Local versions of the Minkowski tensors of convex bodies in \(n\)-dimensional Euclidean space are introduced. An extension of Hadwiger’s characterization theorem for the intrinsic volumes, due to Alesker, states that the continuous, isometry covariant valuations on the space of convex bodies with values in the vector space of symmetric \(p\)-tensors are linear combinations of modified Minkowski tensors. We ask for a local analogue of this characterization, and we prove a classification result for local tensor valuations on polytopes, without a continuity assumption.

Measures on the space of convex bodies

Abstract It is a classical question in convex geometry whether there exists a σ-finite Borel measure on the space of convex bodies such that every open subset has positive measure. In this paper we give a positive answer by introducing a class of measures with the desired feature and additional invariance properties which are (in a certain sense) strongest possible.

A characterization of the duality mapping for convex bodies

We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. The property that the duality interchanges pairwise intersections and convex hulls of unions is sufficient for a characterization, up to a trivial exception and the composition with a linear transformation.

Width of convex bodies in spaces of constant curvature

We consider the measure of points, the measure of lines and the measure of planes intersecting a given convex body K in a space form. We obtain some integral formulas involving the width of K and the curvature of its boundary ∂K. Also we study the special case of constant width. Moreover we obtain a generalisation of the Heintze–Karcher inequality to space forms.

Simple valuations on convex bodies

We determine all continuous translation invariant simple valuations on the space of convex bodies in d-dimensional Euclidean space.

Modelling Clip: Some More Results

AbstractThe modedling clip of the PHIGS ISO Standard is mathematically analysed. The most important result of this analysis is the fact that the projective image of a modding clip body (that is a not necessarily bounded convex body in space) is simply the union of two convex bodies. Furthermore, it will also be proved that in some cases one of these two bodies is empty. This fact makes the implementation of the modelling clip fairly straightforward and makes it also possible to use all already existing results on clipping against general convex bodies without change.

Volume approximation of convex bodies by inscribed polytopes

A convex body C in d-dimensional Euclidean space E ~ is a compact convex subset of lE a with non-empty interior. We say that C is of differentiability class ~k if its boundary considered as a manifold is of class ~k. The symmetric difference metric 6 s, also called Nikodym metric, on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference. Given a convex body C, let ~ = ~ ( C ) (n = d + l, d + 2,...) denote the family of convex polytopes P having at most n vertices and being inscribed into C, that is, all vertices are on the boundary bdC of C. It is well known that for each n there is a polytope P, ~ ~ such that ~s(C, P.) = ~s(C, ~,~)( = inf {~s(C, P): P ~ ~.~}).

The stereology of projected images

SUMMARYA generalized, geometrical approach is adopted in reviewing the basic relationships of projected structures. Projections of features from a plane to a projection line, and from a three‐dimensional foil to a projection plane are considered briefly. The relationships that apply to apparent and total projections are next assembled for lines, surfaces and bounded regions. The properties of convex bodies, the geometrical attributes, and the related projected quantities are enumerated. The inter‐relationships of important microstructural parameters are expressed for convex bodies in space, in sections, and in the projection plane.Based on the important concept of total projection, expressions are obtained that correct for truncation and overlap of particles in foils. Approximations to the total projection are developed for lines obscured by surfaces, and for points hidden by surfaces in space. Finally methods for determining size distributions of particles in thin foils are reviewed and recent modifications to the original procedures are discussed.