  Let $G = (V, E)$ be an edge-weighted geometric graph such that every edge is
horizontal or vertical. The weight of an edge $uv \in E$ is its length. Let $
W_G (u,v)$ denote the length of a shortest path between a pair of vertices $u$
and $v$ in $G$. The graph $G$ is said to be a Manhattan network for a given
point set $ P $ in the plane if $P \subseteq V$ and $\forall p,q \in P$, $ W_G
(p,q)=|pq|_1$. In addition to $ P$, graph $G$ may also include a set $T$ of
Steiner points in its vertex set $V$. In the Manhattan network problem, the
objective is to construct a Manhattan network of small size for a set of $ n $
points. This problem was first considered by Gudmundsson et
al.\cite{gudmundsson2007small}. They give a construction of a Manhattan network
of size $\Theta(n \log n)$ for general point set in the plane. We say a
Manhattan network is planar if it can be embedded in the plane without any edge
crossings. In this paper, we construct a linear size planar Manhattan network
for convex point set in linear time using $\mathcal{ O}(n)$ Steiner points. We
also show that, even for convex point set, the construction in Gudmundsson et
al. \cite{gudmundsson2007small} needs $\Omega (n \log n)$ Steiner points and
the network may not be planar.
