  In the geodetic convexity, a set of vertices $S$ of a graph $G$ is
$\textit{convex}$ if all vertices belonging to any shortest path between two
vertices of $S$ lie in $S$. The cardinality $con(G)$ of a maximum proper convex
set $S$ of $G$ is the $\textit{convexity number}$ of $G$. The
$\textit{complementary prism}$ $G\overline{G}$ of a graph $G$ arises from the
disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a
perfect matching between the corresponding vertices of $G$ and $\overline{G}$.
In this work, we we prove that the decision problem related to the convexity
number is NP-complete even restricted to complementary prisms, we determine
$con(G\overline{G})$ when $G$ is disconnected or $G$ is a cograph, and we
present a lower bound when $diam(G) \neq 3$.
