Related papers: Kissing number in spherical space
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in $\mathbb{H}^n$, for $n\geq 2$. For that purpose, the kissing number is replaced by the kissing function $\kappa(n, r)$ which depends on…
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $\mathbb{H}^n$ and spherical $\mathbb{S}^n$ spaces, for $n\geq 2$. For that purpose, the kissing number is replaced by the…
The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds…
Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions.…
In this paper, we give some new lower bounds for the kissing number of $\ell_p$-spheres. These results improve the previous work due to Xu (2007). Our method is based on coding theory.
The Koebe circle packing theorem states that every finite planar graph can be realized as the nerve of a packing of (non-congruent) circles in R^3. We investigate the average kissing number of finite packings of non-congruent spheres in R^3…
The kissing number of $\mathbb{R}^n$ is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin (2010), based on the semidefinite programming bound of Bachoc…
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps.…
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing…
The average kissing number of $\mathbb{R}^n$ is the supremum of the average degrees of contact graphs of packings of finitely many balls (of any radii) in $\mathbb{R}^n$. We provide an upper bound for the average kissing number based on…
We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular,…
It has been shown that the maximum stable set problem in some infinite graphs, and the kissing number problem in particular, reduces to a minimization problem over the cone of copositive kernels. Optimizing over this infinite dimensional…
In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton's kissing number. This notion has not only led to interesting mathematics, but has also found…
We apply the semidefinite programming approach developed in arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing number in several dimensions where we in…
How many unit $n-$dimensional spheres can simultaneously touch or kiss a central $n-$dimensional unit sphere? Beyond mathematics this question has implications for fields such as cryptography and the structure of biologic and chemical…
In 1694, Gregory and Newton discussed the problem to determine the kissing number of a rigid material ball. This problem and its higher dimensional generalization have been studied by many mathematicians, including Minkowski, van der…
We present an extension of the Delsarte linear programming method. For several dimensions it yields improved upper bounds for kissing numbers and for spherical codes. Musin's recent work on kissing numbers in dimensions three and four can…
A new upper bound $\kappa_T(K_n)\leq 2.9162^{(1+o(1))n}$ for the translative kissing number of the $n$-dimensional cross-polytope $K_n$ is proved, improving on Hadwiger's bound $\kappa_T(K_n)\leq 3^n-1$ from 1957. Furthermore, it is shown…
We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density…
We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar…