2020/04/29 by Wiesław Kubiś, Kubiś, Wiesław, Janusz Morawiec +3
Computer Science · Mathematics · #03E05 #52A10 #52C45 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2004.14301
openalex publication_date 2020/04/29 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study betweenness preserving mappings (we call them 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.