Successive radii and Minkowski addition

In this paper we study the behavior of the so called successive inner and outer radii with respect to the Minkowski addition of convex bodies, generalizing the well-known cases of the diameter, minimal width, circumradius and inradius. We get all possible upper and lower bounds for the radii of the sum of two convex bodies in terms of the sum of the corresponding radii.     

Ein Approximationssatz für konvexe Körper

It is shown that a convex body in n-dimensional Euclidean space can be approximated by a sequence of smooth convex bodies in such a way that the principal radii of curvature converge in a certain sense. This fact is used to characterize those first surface measures of convex bodies which belong to polytopes. Furthermore it is proved that the support function of a convex body whose first surface measure has bounded density must have continuous first partial derivatives.     

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.     

Lp-dual Quermassintegral sums

In this paper,we first introduce a concept of L_p-dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L_p-dual Quermassintegral sums.Moreover,by using Lutwak's width-integral of index i,we establish the L_p-Brunn-Minkowski inequality for the polar mixed projec- tion bodies.As applications,we prove some interrelated results.     

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.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Brunn–Minkowski and Zhang inequalities for convolution bodies

A quantitative version of Minkowski sum, extending the definition of θ$\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn–Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.     

Covering Polygonal Annuli by Strips

In 2000 Bezdek asked which plane convex bodies have the property that whenever an annulus, consisting of the body less a sufficiently small scaled copy of itself, is covered by strips, the sum of the widths of the strips must still be at least the minimal width of the body. We characterise the polygons for which this is so.     

Existence and sum decomposition of vertex polyhedral convex envelopes

Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral.     

A new class of Salagean-type harmonic univalent functions

We define and investigate a new class of Salagean-type harmonic univalent functions. We obtain coefficient conditions, extreme points, distortion bounds, convex combination and radii of convex for the above class of harmonic univalent functions.     

Intrinsic volumes and successive radii

Motivated by a problem of Teissier to bound the intrinsic volumes of a convex body in terms of the inradius and the circumradius of the body, we give upper and lower bounds for the intrinsic volumes of a convex body in terms of the elementary symmetric functions of the so-called successive inner and outer radii. These results improve on former bounds and, in particular, they also provide bounds for the elementary symmetric functions of the roots of Steiner polynomials in terms of the elementary symmetric functions of these radii.     

Convex sub-differential sum rule via convex semi-closed functions with applications in convex programming

This paper deals with some basic notions of convex analysis and convex optimization via convex semi-closed functions. A decoupling-type result and also a sandwich theorem are proved. As a consequence of the sandwich theorem, we get a convex sub-differential sum rule and two separation results. Finally, the derived convex sub-differential sum rule is applied to solving the convex programming problem.     

On quasilinear spaces of convex bodies and intervals

Algebraic systems abstracting properties of convex bodies and intervals, with respect to addition and multiplication by scalars, known as quasilinear spaces, are studied axiomatically. We discuss special quasilinear spaces with group structure called quasivector spaces. We show that every quasivector space is a direct sum of a vector space and a symmetric quasivector space. A complete characterization of symmetric quasivector spaces in the finite dimensional case is given, which permits to reduce computation in quasilinear spaces to computation in familiar vector spaces.     

A characterization of the Euclidean ball in terms of concurrent sections of constant width

We prove that the Euclidean ball is the unique convex body with the property that all its sections through a fixed point are convex bodies of constant width. Furthermore, we characterize those convex bodies which are sections of convex bodies of constant width.Research supported by the Alexander von Humboldt-Foundation.     

Tangible convex bodies. Appendix to “Multivariable Wiener-Hopf Operators I”

Study in a local geometry of non-smooth convex bodies via their supporting cones. The supporting cones are differential objects if the convex bodies are tangible. Examples of completely tangible and non-tangible convex bodies are presented.     

Mixed Polytopes

Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

Mixed Polytopes

   Abstract. Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an inclusion inequality for translation mixtures of convex bodies, where the extremal case characterizes simplices.     

不变的赋值函数与Cauchy投影公式(英文)

本文研究了Rn中凸集上不变的赋值函数与凸体的投影问题.利用赋值函数的方法,我们获得了凸体在任意维平面上投影的Cauchy公式和Kubota公式,这些结果推广了经典的Cauchy公式和Kubota公式.     

The Sine Representation of Centrally Symmetric Convex Bodies

The problem of the sine representation for the support function of centrally symmetric convex bodies is studied. We describe a subclass of centrally symmetric convex bodies which is dense in the class of centrally symmetric convex bodies. Also, we obtain an inversion formula for the sine-transform.     

On fencing problems

Fencing problems regard the bisection of a convex body in a way that some geometric measures are optimized. We introduce the notion of relative diameter and study bisections of centrally symmetric planar convex bodies, bisections by straight line cuts in general planar convex bodies and also bisections by hyperplane cuts for convex bodies in higher dimensions. In the planar case we obtain the best possible lower bound for the ratio between the relative diameter and the area.