关于广义Minkowski安排
On generalized Minkowski arrangements
摘要：
闵可夫斯基排列的概念是由费耶斯·托特在1965年提出的，它是一个中心对称的凸体族，具有这样的性质：族中没有任何成员在其内部包含任何其他成员的中心。这个概念在1967年由Fejes Tóth推广，他称一个中心对称凸体族为一个广义的$μ$阶Minkowski排列，对于某些$0 < μ< 1$，如果族中没有任何成员$K$与任何其他成员$K'$的位似副本重叠，且比率为$μ$，且与$K'$具有相同的中心。本文证明了欧氏平面上多个圆盘的$μ$阶广义Minkowski排列的元素总面积的一个精确上界。这个结果是Fejes Tóth关于圆盘的闵可夫斯基排列的一个类似结果，以及Böröczky和Szabó关于平面上圆盘的广义闵可夫斯基排列的最大密度的一个结果的共同推广。此外，我们给出了中心对称凸体的相似副本的广义Minkowski排列的密度的一个精确的上界。
英文摘要：
The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order $μ$ for some $0 < μ< 1$ if no member $K$ of the family overlaps the homothetic copy of any other member $K'$ with ratio $μ$ and with the same center as $K'$. In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order $μ$ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.
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