On a Lemma of Minkowski

In this paper we show that a generalization of a lemma of Minkowski can be applied to solve two problems concerning kissing numbers of convex bodies. In the first application we give a short proof showing that the lattice kissing number of tetrahedra is eighteen. Moreover, it turns out that for any tetrahedron T there exist a unique 18-neighbour lattice packing of T and an essentially unique 16-neighbour lattice packing of T. Secondly we show that for every integer d ≥ 3 there exists a d-dimensional convex body K such that the difference between its translative kissing number and lattice kissing number is at least 2^d-1.     

Improved Delsarte bounds for spherical codes in small dimensions

We present an extension of the Delsarte linear programming method for spherical codes. For several dimensions it yields improved upper bounds including some new bounds on kissing numbers. Musin's recent work on kissing numbers in dimensions three and four can be formulated in our framework.     

The kissing numbers of tetrahedra

We determine the lattice kissing numbers of tetrahedra, by which we disprove a conjecture by Grünbaum. At the same time, we present a strange phenomenon concerning kissing numbers and packing densities of tetrahedra. This article is based upon part of the author's Ph.D. thesis which was supported by the Austrian Academic Exchange Service.     

An example concerning the translative kissing number of a convex body

This article presents a class of convex bodies inE ^d (d≥3) where their maximum kissing numbers in translative packings are larger than their maximum kissing numbers in lattice packings. This work was supported by the Austrian Academic Exchange Service.     

Some remarks concerning kissing numbers,blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3 ^d –1.     

A Note on the Kissing Numbers of Superballs

The lower bounds for the translative kissing numbers of superballs are studied in this note. We improve the bound given by Larman and Zong. Furthermore, we give a constructive bound based on algebraic-geometry codes that also improves the bound by Larman and Zong in almost all cases.     

Bounds for codes by semidefinite programming

Delsarte’s method and its extensions allow one to consider the upper bound problem for codes in two-point homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that using as variables power sums of distances, this problem can be considered as a finite semidefinite programming problem. This method allows one to improve some linear programming upper bounds. In particular, we obtain new bounds of one-sided kissing numbers.     

Functionals on the spaces of convex bodies

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.     

The Translative Kissing Number of Cartesian Product of two Convex Bodies, one of which is Two-Dimensional

This article discusses the relation between the translative kissing numbers of convex bodies K ₁, K ₂ and K ₁ K ₂. As an application of the main theorem, we find that the translative kissing number of B Q, where B is a 24-dimensional ball and Q is a two-dimensional non-parallelogram convex domain, is 1375926.     

On Certain Coxeter Lattices Without Perfect Sections

In this paper, we compute the kissing numbers of the sections of the Coxeter lattices , n odd, and in particular we prove that for n 7 they cannot be perfect. The proof is merely combinatorial and relies on the structure of graphs canonically attached to the sections.     

Octahedral designs

An oriented octahedral design of order v, or OCT(v), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT(v) exists if and only if v≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT(v) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT(v) with a subdesign OCT(U) exists if and only if v and u are admissible and v≥u+4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010     

On the extremality of the polynomials used for obtaining the best known upper bounds for the kissing numbers

We prove that the polynomials used for obtaining the best known upper bounds for some kissing numbers (the maximum number of nonoverlapping unit spheres that can touch a unit sphere in n dimensions) are best between the polynomials of the same or lower degree. We give also some extremal polynomials we have obtained using a method proposed in [4]. The upper bounds obtained in this way are slightly better than these from [1]. However the improvements are not in the integer part for dimensionsn 18.     

Improved Linear Programming Bounds for Antipodal Spherical Codes

   Abstract. Let S\subset[-1,1) . A finite set \Ccal=\set x _{i i=1} ^M \subset\Re ⁿ is called a spherical S-code if \norm x _i =1 for each i , and x _i \tran x _j ∈ S , i\ne j . For S=[-1, 0.5] maximizing M=|C| is commonly referred to as the kissing number problem. A well-known technique based on harmonic analysis and linear programming can be used to bound M . We consider a modification of the bounding procedure that is applicable to antipodal codes; that is, codes where x∈\Ccal\implies -x∈\Ccal . Such codes correspond to packings of lines in the unit sphere, and include all codes obtained as the collection of minimal vectors in a lattice. We obtain improvements in upper bounds for kissing numbers attainable by antipodal codes in dimensions 16≤ n≤ 23 . We also show that for n=4 , 6 and 7 the antipodal codes with maximal kissing numbers are essentially unique, and correspond to the minimal vectors in the laminated lattices \Lam _n .     

HIGHER ORDER FLEXIBILITY OF OCTAHEDRA

More than hundred years ago R. Bricard determined all continuously flexible octahedra. On the other hand, also the geometric characterization of first-order flexible octahedra has been well known for a long time. The objective of this paper is to analyze the cases between, i.e., octahedra which are infinitesimally flexible of order n > 1 but not continuously flexible. We prove explicit necessary and sufficient conditions for the orders two, three and even for all n < 8, provided the octahedron under consideration is not totally flat. Any order ≥ 8 implies already continuous flexibility, as the configuration problem for octahedra is of degree eight. This revised version was published online in August 2006 with corrections to the Cover Date.     

Some Applications of the Formula for the Volume of an Octahedron

We introduce the notions of combinatorial, metric and spatial symmetries of a polyhedron. In the case of symmetric octahedra, we present explicit forms of canonical polynomials for determining their volumes.     

An application of kissing number in sum-product estimates

The boundedness of the kissing numbers of convex bodies has been known to Hadwiger [9] for long. We present an application of it to the sum-product estimate

$$\max(\mid{\mathcal{A}+\mathcal{A}}\mid,\mid{\mathcal{A}\mathcal{A}}\mid)\gg \frac {\mid{\mathcal{A}\mid}^{4/3}}{\lceil\log\mid{\mathcal{A}\mid}\rceil^{1/3}}$$

for finite sets \({\mathcal{A}}\) of quaternions and of a certain family of well-conditioned matrices.

     

The Kissing Problem in Three Dimensions

The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. The first proof that k(3) = 12 was given by Schutte and van der Waerden only in 1953. In this paper we present a new solution of the Newton--Gregory problem that uses our extension of the Delsarte method. This proof relies on basic calculus and simple spherical geometry.     

Isodual Codes over ℤ2k and Isodual Lattices

A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over _2k . These codes give rise to isodual lattices; in particular, we construct a 22-dimensional isodual lattice with minimum norm 3 and kissing number 2464.     

Lattice packing and covering of convex bodies

The aim of this article is twofold. First, to indicate briefly major problems and developments dealing with lattice packings and coverings of balls and convex bodies. Second, to survey more recent results on uniqueness of lattice packings and coverings of extreme density, on characterization of local minima and maxima of the density and on estimates of the kissing number. Emphasis is on results in general dimensions.