Chengfei Xie, Gennian Ge
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.
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A kissing sphere is a sphere that is tangent to a fixed reference ball. We develop in this paper a distance geometry for kissing spheres, which turns out to be a generalization of the classical Euclidean distance geometry.
Let H be a closed half-space of n-dimensional Euclidean space. Suppose S is a unit sphere in H that touches the supporting hyperplane of H. The one-sided kissing number B(n) is the maximal number of unit nonoverlapping spheres in H that can touch S. Clearly, B(2)=4. It was proved that B(3)=9. Recently, K. Bezdek proved that B(4)=18 or 19, and conjectured that B(4)=18. We present a proof of this conjecture.
In 1969, Fejes Toth conjectured that in Euclidean 3-space any packing of equal balls such that each ball is touched by twelve others consists of hexagonal layers. This article verifies this conjecture.
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 for spherical codes and sphere packings. We survey old and recent results on the kissing numbers keeping the generality of spherical codes.
Average kissing numbers for non-congruent sphere packings
柯比圈填充定理指出，每个有限可平面图都可以被实现为R^3中（非全等）圈的填充的神经。我们研究了R^3中非全等球面有限填充的平均吻数，作为对这种填充可能的神经的第一个限制。我们证明了所有填充的平均接吻数的上确界k满足12.566 ~ 666/53 <= k < 8 + 4*sqrt（3）~ 14.928.我们利用资源耗尽论证得到了上界的上界，并利用S^3中包含球冠填充的构造得到了上界的上界.我们的结果与两个关于平均接吻数的天真假设相矛盾：它是无界的，或者它是由全等球面的无限堆积而最大化的。
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 particular our computations give the (known) values for the cases n = 3, 4, 8, 24.
This article sketches the proofs of two theorems about sphere packings in Euclidean 3-space. The first is K. Bezdek's strong dodecahedral conjecture: the surface area of every bounded Voronoi cell in a packing of balls of radius 1 is at least that of a regular dodecahedron of inradius 1. The second theorem is L. Fejes Toth's contact conjecture, which asserts that in 3-space, any packing of congruent balls such that each ball is touched by twelve others consists of hexagonal layers. Both proofs are computer assisted. Complete proofs of these theorems appear in the author's book "Dense Sphere Pa…
The kissing number problem asks for the maximal number k(n) of equal size nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Schütte and van der Waerden. In this paper we present a solution of a long-standing problem about the kissing number in four dimensions. Namely, the equality k(4)=24 is proved. The proof is based on a modification of Delsarte's method.
Variations on five-dimensional sphere packings
本文分析了Szöllősi最近提出的猜想最优五维 kissing 配置，并构造了新的配置，扩展了Conway和Sloane的猜想最优堆叠列表，同时提出了新的九维 kissing 配置。
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 and Vallentin (2008), computed the best known upper bounds for the kissing number for several values of $n \leq 23$. In this paper, we exploit the symmetry present in the semidefinite programming bound to provide improved upper bounds for $n = 9, \ldots, 23$.
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.
This paper investigates the behaviour of the kissing number $κ(n, r)$ of congruent radius $r > 0$ spheres in $\mathbb{S}^n$, for $n\geq 2$. Such a quantity depends on the radius $r$, and we plot the approximate graph of $κ(n, r)$ with relatively high accuracy by using new upper and lower bounds that are produced via semidefinite programming and by using spherical codes, respectively.
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 $κ(n, r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $κ(n, r)$, we study their asymptotic behaviour and show, in particular, that $\lim_{r\to \infty} \frac{\log κ(n,r)}{r} = n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.
Improved kissing numbers in seventeen through twenty-one dimensions
研究改进了十七至二十一维的 kissing 数，通过调整晶格向量符号来增加额外球体的空间，提升了各维度的 kissing 数。
In this paper we consider the following topics related to results of László Fejes Tóth: (1) The Tammes problem and Fejes Tóth's bound on circle packings; (2) Fejes Tóth's problem on maximizing the minimum distance between antipodal pairs of points on the sphere; (3) Fejes Tóth's problem on the maximum kissing number of packings on the sphere; (4) The Fejes Tóth -- Sachs problem on the one--sided kissing numbers; (5) Fejes Tóth's papers on the isoperimetric problem for polyhedra.
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 macromolecules. The kissing number is only known for dimensions 1-4, 8 and 24 (2, 6, 12, 24, 240, 19650, respectively) and only particularly obvious for dimensions one and two. Indeed, in four dimensions it is not even known if Platonic polytope unique to that dimension known as the 24-cell is the unique kissing configuration. We have not been able to prove that …
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…
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 number based on semidefinite programming. This paper is a report on high accuracy calculations of these upper bounds for n <= 24. The bound for n = 16 implies a conjecture of Conway and Sloane: There is no 16-dimensional periodic point set with average theta series 1 + 7680q^3 + 4320q^4 + 276480q^5 + 61440q^6 + ...
We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a lattice to a wide class of test functions. From this we obtain quite general transference bounds, as well as bounds on the number of lattice points contained in certain bodies. As applications, we bound the lattice kissing number in $\ell_p$ norms by $e^{(n+ o(n))/p}$ for $0 < p \leq 2$, and also give a proof of a new transference bound in the $\ell_1$ norm.
There has been recent work using Shape Theory to answer the longstanding and conceptually interesting problem of what is the probability that a triangle is obtuse. This is resolved by three kissing cap-circles of rightness being realized on the shape sphere; integrating up the interiors of these caps readily yields the answer to be 3/4. We now generalize this approach by viewing rightness as a particular value of maximal angle, and then covering the shape sphere with the maximal angle flow. Therein, we discover that the kissing cap-circles of rightness constitute a separatrix. The two qualitat…
Finding Kissing Numbers with Game-theoretic Reinforcement Learning
将亲吻数问题转化为合作矩阵补全博弈，利用强化学习系统PackingStar在极值配置空间中探索，改进了15个长期未突破的亲吻数上界，并发现了新的可解释几何结构。
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 $δ(\mathcal{P})$ of an arbitrary infinite non-congruent sphere packing $\mathcal{P}$.
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter-Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter-Todd lattice is analogous to the…
I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsarte's method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed t…
This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two long-standing open problems: the uniqueness of maximum kissing arrangements in 4 dimensions and the 24-cell conjecture. Note that a proof of the 24-cell conjecture also proves that the checkerboard lattice packing D4 is the densest sphere packing in 4 dimensions.
Non-convex polytopes realizing kissing contact graphs in dimensions 5-7
本文研究5-7维中九个猜想最优非格点亲吻构型，证明它们均为单位边多胞形顶点集，其中八个非凸，通过分裂格点接触多胞形面构造，并定义折痕多胞形类别，所有陈述经精确算术验证。
Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.
We use the automorphism group $Aut(H)$, of holes in the lattice $L_8=A_2\oplus A_2\oplus D_4$, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, $L_4=A_2\oplus A_2$, enters the construction because a subgroup of $Aut(L_4)$ is isomorphic to $Aut(H)$. The lattices $L_8$ and $L_4$, when glued together through this relationship, provide an alternative construction of the laminated lattice in twelve dimensions with kissing number 648. More interestingly, the action of $Aut(H)$ on $L_4$ defines a pair of invariant planes through which dense, non-…
Lattices with Large Kissing Number from Local Density
针对具有大吻数的局部密度格，证明了相关下界并改进了此前的最优结果，通过构造精确局部密格得到了大吻数格的高效随机构造方法。
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 applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers o…
Solving clustered low-rank semidefinite programs arising from polynomial optimization
针对源自多项式优化的聚类低秩半定规划高精度求解需求，提出结合采样、对称性约简与贪心基选取的专用原-对偶内点法，在接吻数界计算等问题上实现加速且对病态问题表现良好。
Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100-year-old lower bound on the maximal packing density due to Minkowski in $d$-dimensional Euclidean space $\Re^d$. The asymptotic behavior of this bound is controlled by $2^{-d}$ in high dimensions. Using an optimization procedure that we introduced earlier \cite{To02c} and a conjecture concerning the existence of disordered sphere packings in $\Re^d$, we obtain a provisional lower bound on th…
On $\ell_1$ embeddings of finite metric spaces, and sphere-of-influence graphs
本文研究了有限度量空间的ℓ₁嵌入问题，提出了配对切割锥的概念，并探讨了影响球图的构造与性质。
Optimal Extensions of Cross-Sections: Sphere Packings in Dimensions 38 to 43
该研究在39维和43维构造非格球体填充，通过对映体构造应用于48维晶格横截面实现，改进了记录，43维填充提高了接吻数，39维填充有新的十点对映体簇，且提供了机器可检查证书等。
NP-completeness of the $\ell_1$-embedding problem for simple graphs as sphere-of-influence graphs
本文证明了简单图作为影响球图的ℓ₁嵌入问题是一个NP完全问题。
The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings
综述分析中逐项保持半正定性的函数问题，重点讨论无维情形，并联系距离几何、度量嵌入、正定序列与函数、傅里叶分析、协方差估计、Schur多项式及有限域，附录介绍球体堆积与亲吻数。
The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random polytopes have uniformly absolutely bounded isotropic constants with overwhelming probability. This generalizes the result for the Euclidean sphere ($p=2$) obtained by D. Alonso-Gutiérrez. The proof requires several different tools including a probabilistic representation of the cone measure due to G. Schechtman and J. Zinn and moment estimates for sums of inde…
Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid
本文提出了一种在高维空间中通过随机演化的椭球构造高密度球体排列的方法，证明了其密度下限为c n²·2⁻ⁿ，优于之前的结果。
On the uniform continuity of homeomorphisms between the spheres of $\ell_\infty^k$ and $\ell_1^k$
研究了$\ell_\infty^k$和$\ell_1^k$球面之间同胚的均匀连续性问题，证明了在特定条件下无法存在等距连续的同胚序列，并提供了相关估计和不等式。
From the separable Tammes problem to extremal distributions of great circles in the unit sphere
本文研究单位球面上全等球冠的完全可分离堆积与覆盖问题，解决了最多8个球冠的可分离Tammes问题，证明三维欧氏球的中心可分离亲吻数为8，并给出大圆铺砌胞腔最小内切半径及TS-覆盖的界，部分结果推广到高维球面。
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