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 Schütte 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.
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 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 semidefinite programming that improves previous bounds in dimensions $3, \ldots, 9$. A very simple upper bound for the average kissing number is twice the kissing number; in dimensions $6, \ldots, 9$ our new bound is the first to improve on this simple upper bound.
Improved kissing numbers in seventeen through twenty-one dimensions
Microsoft Research New England（微软研究院新英格兰）；Department of Mathematics Massachusetts Institute of Technology（麻省理工学院数学系）；Department of Mathematics Stanford University（斯坦福大学数学系）；Department of Computer Science University of Washington（华盛顿大学计算机科学系）
László Fejes Tóth and Aladár Heppes proposed the following generalization of the kissing number problem. Given a ball in $\mathbb{R}^d$, consider a family of balls touching it, and another family of balls touching the first family. Find the maximal possible number of balls in this arrangement, provided that no two balls intersect by interiors, and all balls are congruent. They showed that the answer for disks on the plane is $19$. They also conjectured that if there are three families of disks instead of two, the answer is $37$. In this paper we confirm this conjecture.
In this article we construct a sequence $\{M_i\}$ of non compact finite volume hyperbolic $3$-manifolds whose kissing number grows at least as $\mathrm{vol}(M_i)^{\frac{31}{27}-ε}$ for any $ε>0$. This extends a previous result due to Schmutz in dimension $2$.
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 kissing function $κ_H(n, r)$, resp. $κ_S(n, r)$, which depends on the dimension $n$ and the radius $r$. After we obtain some theoretical upper and lower bounds for $κ_H(n, r)$, we study their asymptotic behaviour and show, in particular, that $κ_H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac{n-1}{2}, \frac{1}{2}) \cdot e^{(n-1) r}$, where $d_n$ is the sphere pa…
Lattices with Large Kissing Number from Local Density
University of Colorado Boulder（科罗拉多大学博尔德分校）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；National University of Singapore（新加坡国立大学）；Cornell University（康奈尔大学）
Microsoft Research New England（微软研究院新英格兰）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）；Massachusetts Institute of Technology（麻省理工学院）
A lattice family with kissing numbers $τ(\mathcal{L}_n) \ge e^{2 \sqrt{n}}$
NXP Semiconductors（恩智浦半导体）；Harvard Business School（哈佛商学院）；Center of Mathematical Sciences and Applications, Harvard University（哈佛大学数学科学与工程应用中心）；Technological Innovation, Entrepreneurship, and Strategic Management (TIES) Group at the MIT Sloan School of Management（麻省理工学院斯隆管理学院技术创新、创业与战略管理（TIES）小组）
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（处理器重点实验室，中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
