论文标题
在本地可分离单位球形包装的接触数量上
On contact numbers of locally separable unit sphere packings
论文作者
论文摘要
欧几里得$ d $空间有限的许多球包装的接触数是包装中的接触对球的数量。所谓的完全可分开的球形包装形成了一个突出的球体包装的亚科：在这里，如果可以用超平面将任何两个球隔开，以使得与包装中每个球的室内隔离不相交，则在欧几里得$ d $ -space中的一堆球被称为完全可分离。 Bezdek，Szalkai和Szalkai（离散数学339（2）：668-676，2016）上层以$ n $ n $和$ d $表示的完全可分离的$ n $单位球的完全可分离包装的接触数量。在本文中，我们改善了它们的上限，并将新的上限扩展到所谓的本地球的本地可分离包装。如果包装的每个单位球与与之相切的单位球一起形成完全可分开的包装，我们将单位球包装为本地可分开的包装。在飞机上，我们通过表征所有具有最大联系数的$ n $单位磁盘的本地可分离包装来证明结晶的结果。
The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of balls in Euclidean $d$-space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) upper bounded the contact numbers of totally separable packings of $n$ unit balls in Euclidean $d$-space in terms of $n$ and $d$. In this paper we improve their upper bound and extend that new upper bound to the so-called locally separable packings of unit balls. We call a packing of unit balls a locally separable packing if each unit ball of the packing together with the unit balls that are tangent to it form a totally separable packing. In the plane, we prove a crystallization result by characterizing all locally separable packings of $n$ unit disks having maximum contact number.
