  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
Chv\'{a}tal asked whether an analogous statement holds within the framework of
finite metric spaces, with lines defined using the notion of {\em betweenness}.
  In this paper, we prove that in the plane with the $L_1$ (also called
Manhattan) metric, a non-collinear set of $n$ points induces at least $\lceil
n/2\rceil$ 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 $L_{\infty}$
metric.
