AI 中文总结
本文研究三维空间中n个全等球堆积的最大接触数c(n)，证明其极小刚性，确定n=6至9时的精确值，并给出一般下界及面心立方晶格情形下的上界与渐近公式。
AI 中文摘要
在三维欧氏空间\mathbb{R}^3中，n个全等球的堆积是一族具有相同半径且内部互不相交的欧氏球。堆积的接触数是指球之间接触对的数量。本文研究确定\mathbb{R}^3中n个全等球堆积的最大接触数c(n)的问题。我们首先证明所有具有接触数c(n)的n个全等球堆积都是极小刚性的。进一步，我们证明对于n=6,7,8,9，有c(n)=3n-6。这两个结果解决了K. Bezdek和Khan的一个猜想。在证明后一个结果的过程中，我们还枚举了n=6,7,8时所有具有接触数c(n)的n个全等球堆积的接触结构。此外，我们提供了一个下界构造，表明当n=16k^3-33k^2+24k-6（其中k∈\mathbb{N}）时，c(n)>6n-6\sqrt[3]{2}n^{\frac{2}{3}}。我们还研究了每个球心位于面心立方晶格A_3上的受限问题。在这种情况下，令c_{A}(n)表示最大接触数。我们证明对所有n，有c_{A}(n)≤6n-\frac{6}{\sqrt[6]{2}}n^{\frac{2}{3}}，并确定c_{A}(n)的渐近行为为c_{A}(n)=6n-(1+o(1))6\sqrt[3]{2}n^{\frac{2}{3}}。
英文摘要
A packing of $n$ congruent balls in $\mathbb{R}^3$ is a family of interior-disjoint Euclidean balls all having the same radius. The contact number of a packing is the number of touching pairs of balls. In this paper we investigate the problem of determining the maximum contact number, $c(n)$, of a packing of $n$ congruent balls in $\mathbb{R}^3$. We first show that all packings of $n$ congruent balls that have a contact number of $c(n)$ are minimally rigid. Furthermore, we show that $c(n)=3n-6$ for $n=6,7,8,$ and $9$. These two results resolve a conjecture of K. Bezdek and Khan. During the proof of the latter result, we also enumerate the contact structures of all packings of $n$ congruent balls with contact number $c(n)$ for $n=6,7,$ and $8$. Additionally, we provide a lower bound construction which shows $c(n)> 6n-6\sqrt[3]{2}n^\frac{2}{3}$ when $n=16k^3-33k^2+24k-6$ where $k\in \mathbb{N}$. We also look at the restricted problem where each ball is centered on the face-centered cubic lattice $A_3$. In this case let $c_{A}(n)$ denote the maximum contact number. We show that $c_{A}(n)\leq 6n-\frac{6}{\sqrt[6]{2}}n^\frac{2}{3}$ for all $n$, and determine the asymptotics of $c_{A}(n)$ to be $c_{A}(n)=6n-(1+o(1))6\sqrt[3]{2}n^\frac{2}{3}$.
