欧氏空间和其它赋范空间中的带图
Zone Diagrams in Euclidean Spaces and in Other Normed Spaces
摘要：
区域图是沃罗诺伊图的经典概念的变体。给定度量空间中的n个竞争领土的站点，区域图是竞争中的平衡状态。形式上，它被定义为某个“支配”地图的固定点。Asano，Matousek和Tokuyama证明了欧氏平面上点的区域图的存在性和唯一性，Reem和Reich证明了任意度量空间中两个任意点的存在性。我们建立的存在性和唯一性n不相交的紧网站在欧氏空间的任意（有限）维，更一般地说，在有限维赋范空间的光滑和圆形规范。的证明是相当简单的比浅野等人。我们还提供了一个例子的非唯一性的规范是圆形的，但不光滑。最后，我们证明了存在性和唯一性的两个点的网站在平面上的光滑（但不一定是圆形）的规范。
英文摘要：
Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.
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。
