  A well-known theorem of in plane geometry states that any set of $n$
non-collinear points in the plane determines at least $n$ lines. Chen and
Chv\'atal 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 $L_1$ (also called
Manhattan) metric, a non-collinear set induces at least $\lceil n/2\rceil$
lines. This is an improvement of the previous lower bound of $n/37$, with
substantially different proof.
