Zone diagrams in Euclidean space and other normed spaces
Zone diagrams in Euclidean space and other normed spaces
复制标题
欧几里得空间和其他赋范空间中的区域图
DOI：
10.1007/s00208-011-0761-1
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
T.Tokuyama
中科院分区：
文献类型：
--
作者：
A.Kawamura;J.Matousek;T.Tokuyama
区域图是经典的Voronoi图概念的一种变体。在度量空间中，若有竞争地盘的站点，则区域图是竞争中的一个均衡状态。形式上，它被定义为某个“支配”地图的固定点。浅野，Matoušek和Tokuyama证明了存在性和唯一性的区域图点网站在欧几里德平面，和Reem和赖希表明存在两个任意网站在任意度量空间。我们建立存在性和唯一性forndisjoint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。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范数是圆的，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。
Zone diagrams are a variation on the classical concept of Voronoi diagrams. Givennsites 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, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness forndisjoint 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.