Abstract
A well-known theorem in plane geometry states that any set of n non-collinear points in the plane determines at least n lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of betweenness.
In this paper, we prove that in the plane with the L1 (also called Manhattan) metric, a non-collinear set of n points induces at least LEFT CEILING n/ 2 RIGHT CEILING lines. This is an improvement of the previous lower bound of n/37, with substantially different proof.
As a consequence, we also get the same lower bound for non-collinear point sets in the plane with the Loo metric.