论文标题
单调映射和线条
Monotone mappings and lines
论文作者
论文摘要
我们研究了保留映射（我们称它们为\ emph {monotone}）定义在平面子集上的映射。域是凸组集合后，这样的映射要么是同型的限制，要么其图像包含在一条线和单点的联合中，或者其图像由五个点组成，其中一个是在其他四个点的两个不相交对之间。我们还表明，一个开放式平面套件无法以一对一的单调方式映射到真实的线上。从中，我们推断出，从具有非空内部装置的凸面平面映射一对一的单调映射必然是部分同谱。最后，我们证明了由三个成对非平行线组成的集合不接受一对一的单调映射到真实线中，而另一方面，由单个点相交的三个封闭线段组成的集合确实接纳了这样的映射。
We study betweenness preserving mappings (we call them \emph{monotone}) defined on subsets of the plane. Once the domain is a convex set, such a mapping is either the restriction of a homography, or its image is contained in the union of a line and a single point, or its image consists of five points, one of them being between two disjoint pairs of the other four points. We also show that an open planar set cannot be mapped in a one-to-one monotone way into the real line. From this we deduce that a one-to-one monotone mapping from a convex planar set with nonempty interior is necessarily a partial homography. Finally, we prove that a set consisting of three pairwise non-parallel lines does not admit a one-to-one monotone mapping into the real line, while on the other hand a set consisting of three closed line segments intersecting at a single point does admit such a mapping.
