arXiv math/05110712005-11math.MGmath.CO

The one-sided kissing number in four dimensions

Oleg R. Musin

发表信息：: Periodica Math. Hungarica, vol. 53, No. 1-2 (September 2006), pp. 209-225

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

An extension of Delsarte's method. The kissing problem in three and four dimensions

These lecture notes treat the solution of the kissing number problem in four dimesions which is based on an extension of the Delsarte method for spherical codes.

2005-12-30math.COmath.MG

Five Essays on the Geometry of László Fejes Tóth

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.

2017-07-09math.COmath.MG

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.

2006-10-04math.MG

论凸体的广义吻接数（II）

On Generalized Kissing Numbers of Convex Bodies (II)

本文作为凸体广义吻接数研究项目的延续，得出了3至8维球体在特定参数下的精确广义吻接数，并确定了四维正轴体的格点吻接数。

2025-01-12math.COmath.MG

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 …

2013-01-21math.MGphysics.comp-ph

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。

2024-07-24math.MG

更多相关论文收起更多论文

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…

2009-01-08math.ATmath.COmath.MG

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…

2012-04-06math.MG

十九维空间中 kissing 数的新下界

A new lower bound for the kissing number in 19 dimensions

National University of Singapore（新加坡国立大学）

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。

2026-03-20math.COmath.MG

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^+。

2025-02-05math.MG

五维球体堆叠的变体

Variations on five-dimensional sphere packings

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）

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。

2026-03-04math.COmath.MG

New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

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…

2006-03-05math-phmath.MGmath.MP

逐项演算与无维正性保持者，附球体堆积附录

The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

Indian Institute of Science（印度科学学院）

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。

2026-03-24math.CAmath.FAmath.MG