A short solution of the kissing number problem in dimension three

十七至二十一维中的改进 kissing 数

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。

New upper bounds for kissing numbers from semidefinite programming

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.

The kissing number in four dimensions

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.

三维空间中的接触多胞体

Kissing polytopes in dimension 3

该研究确定了三维立方体中两个不相交格多胞体的最小距离公式，通过建模为格点最小化问题并利用符号计算求解多项式根来证明。

Delimiting Maximal Kissing Configurations in Four Dimensions

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 …

Improved Lower Bounds for Kissing Numbers in Dimensions 25 Through 31

The best previous lower bounds for kissing numbers in dimensions 25 through 31 were constructed using a set $S$ with $|S| = 480$ of minimal vectors of the Leech Lattice, $Λ_{24}$, such that $\langle x, y \rangle \leq 1$ for any distinct $x, y \in S$. Then, a probabilistic argument based on applying automorphisms of $Λ_{24}$ gives more disjoint sets $S_i$ of minimal vectors of $Λ_{24}$ with the same property. Cohn, Jiao, Kumar, and Torquato proved that these subsets give kissing configurations in dimensions 25 through 31 of given size linear in the sizes of the subsets. We achieve $|S| = 488$ b…

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 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.

A survey on the kissing numbers

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.

Semidefinite programming bounds for the average kissing number

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.

关于凸体的广义接触数

On Generalized Kissing Numbers of Convex Bodies

本文引入并研究了凸体接触数的进一步推广，获得了精确结果，特别是针对三、四和八维球体。

Rigidity of spherical codes

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…

十九维空间中 kissing 数的新下界

A new lower bound for the kissing number in 19 dimensions

本文通过奇符号构造方法，证明十九维 kissing 数至少为11948，改进了Cohn和Li的下界。通过构造一个特定的二进制码，得出了1280字的码长。

5-7维亲吻构型上的非凸单位边多胞形

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。

High accuracy semidefinite programming bounds for kissing numbers

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 + ...

从码构造具有指数接吻数的格的困难

Difficulties Constructing Lattices with Exponential Kissing Number from Codes

本文通过反例指出从纠错码构造指数接吻数格的多种自然方法无法保持最短向量对应，并证明 Vlăduţ 相关论文及2024年预印本结论无效，该构造问题仍开放。

Hyperbolic 3-manifolds with large kissing number

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 $2$.

关于交叉多胞体接吻数的研究

On the kissing number of the cross-polytope

本文研究 n 维交叉多胞体的接吻数问题，证明了平移接吻数的新上界 2.9162^{(1+o(1))n} 和新下界 1.1637^{(1-o(1))n}，并确定了四维情形下唯一达到最大格接吻数 40 的格为 D_4^+。

Kissing number in non-Euclidean spaces of constant sectional curvature

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…

具有指数大亲吻数的格

Lattices with exponentially large kissing numbers

本文构造了一列维数趋于无穷的格，其亲吻数呈指数增长，并给出最大格亲吻数的渐近下界。

具有亲吻常数 $\tau(\mathcal{L}_n) \ge e^{\sqrt{n}}$ 的格族

A lattice family with kissing numbers $τ(\mathcal{L}_n) \ge e^{2 \sqrt{n}}$

本文提出一种素数维度整数格族，其亲吻常数下界达到 $e^{\sqrt{n}}$ 量级，优于 Barnes-Wall 格，基于 Reed-Solomon 码推广构造。

Positive definite functions in distance geometry

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…

具有大吻数的局部密度格

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。

Laminating lattices with symmetrical glue

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-…

Towards a proof of the 24-cell conjecture

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.

Codes in spherical caps

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. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered. It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.

求解源自多项式优化的聚类低秩半定规划问题

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。

Functionals on the Spaces of n-Dimensional 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…

Contact numbers for sphere packings

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…