2022/09/20 by Gabriela Araujo‐Pardo, Araujo-Pardo, Gabriela, Martı́n Matamala +3
Mathematics · Physics and Astronomy · #05B #Advanced Differential Geometry Research #Combinatorics (math.CO) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2209.10028
openalex publication_date 2022/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of n non-collinear points in the Euclidean plane defines at least n different lines. Chen and Chvátal in 2008 conjectured that the same results is true in metric spaces for an adequate definition of line. More recently, this conjecture was studied in the context of quasi-metric spaces. In this work we prove that there is a quasi-metric space on four points a, b, c and d whose betweenness is B=(c,a,b),(a,b,c),(d,b,a),(b,a,d). Then, this space has only three lines none of which has four points. Moreover, we show that the betweenness of any quasi-metric space on four points with this property is isomorphic to B. Since B is not metric, we get that Chen and Chvátal's conjecture is valid for any metric space on four points.