赋范空间中平分线的有界表示与径向投影
Bounded representation and radial projections of bisectors in normed spaces
摘要：
众所周知，赋范空间中平分线的拓扑和几何性质的描述是一个非平凡的课题。本文在有限维实巴拿赫空间中引入了平分线的有界表示的概念。这个有用的概念结合了单位球的平分线和阴影边界的概念，两者对应于相同的空间方向。有界表示可视化的二分法和阴影边界（引理1）的拓扑结构之间的连接，并给出了可能性，以简化和扩展一些已知的二分法的径向投影的结果。我们的主要结果（定理1）说，在流形的情况下，闭平分线的拓扑和它的有界表示的拓扑是相同的;它们是以标准方式嵌入欧几里得$n$ -空间中的闭的、$(n-1)$ -维球。
英文摘要：
It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow boundary of the unit ball, both corresponding with the same spatial direction. The bounded representation visualizes the connection between the topology of bisectors and shadow boundaries (Lemma 1) and gives the possibility to simplify and to extend some known results on radial projections of bisectors. Our main result (Theorem 1) says that in the manifold case the topology of the closed bisector and the topology of its bounded representation are the same; they are closed, $(n-1)$-dimensional balls embedded in Euclidean $n$-space in the standard way.
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