抽象空间中的区域图和双区域图
Zone and double zone diagrams in abstract spaces
摘要：
区域图是一个相对较新的概念，它首先由T. Asano，J. Matousek和T.德山它可以被解释为几个相互敌对的王国之间的平衡状态。形式上，它是某个映射的一个不动点。这些作者认为欧几里德平面和证明存在性和唯一性的区域图。在本文中，我们以各种方式推广这一概念。考虑m-空间（度量空间的一个简单推广）中的一般点，并证明了在此情形下的几个存在性和（非）唯一性结果。与以前的作品相比，我们（相当简单）的证明是基于纯粹的顺序理论的论点。给出了许多明确的例子，其中一些例子说明了在一般情况下发生的新现象。我们还重新解释区域图作为一个稳定的配置在一定的组合游戏，并提供了一个算法，在特定的情况下找到这种配置。
英文摘要：
A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast to previous works, our (rather simple) proofs are based on purely order theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.
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。
