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 et al. (Discrete Math. 339(2) (2016), 668–676) 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.

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关于本地可分离单元球填料的联系号码

在欧几里得d空间中有限多个球的包装的接触数是包装中接触的球对的数量。一个突出的球体填充亚族是由所谓的完全可分离的球体填充形成的：这里，如果任何两个球可以被一个超平面分开使得它与球体不相交，那么欧几里得d空间中的球体填充被称为完全可分离的。包装中每个球的内部。贝兹德克等人。( Discrete Math . 339(2) (2016), 668–676)根据n和d对欧几里得d空间中n 个单位球的完全可分离填料的接触数进行了上限. 在本文中，我们改进了它们的上限并将新的上限扩展到所谓的单位球的局部可分离填充。如果填料的每个单元球与与其相切的单元球一起形成完全可分离的填料，我们将单元球填料称为局部可分离填料。在平面上，我们通过表征具有最大接触数的n 个单位圆盘的所有局部可分离填料来证明结晶结果。