  Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in
$D$ if, for every pair of vertices $x, y \in S$, the vertex set of every
$xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is
contained in $S$. The convexity number, ${\rm con}(D)$, of a non-trivial
oriented graph, $D$, is the maximum cardinality of a proper convex set of $D$.
The strong convexity spectrum of the graph $G$, $S_{SC} (G)$, is the set $\{{
\rm con}(D) \colon\ D {\rm \ is \ a \ strong \ orientation \ of \ } G \}$. In
this paper we prove that the problem of determining the convexity number of an
oriented graph is $\mathcal{NP}$-complete, even for bipartite oriented graphs
of arbitrary large girth, extending previous known results for graphs. We also
determine $S_{SC} (P_n \Box P_m)$, for every pair of integers $n,m \ge 2$.
