Consider a convex domain $B$ of $\mathbbR^3$ . We prove that
there exist complete minimal surfaces that are properly immersed
in $B$ . We also demonstrate that if $D$ and $D'$ are convex
domains with $D$ bounded and the closure of $D$ contained in $D'$ ,
then any minimal disk whose boundary lies in the boundary of $D$
can be approximated in any compact subdomain of $D$ by a complete
minimal disk that is proper in $D'$ . We apply these results to
study the so-called type problem for a minimal surface: we
demonstrate that the interior of any convex region of
$\mathbbR^3$ is not a universal region for minimal surfaces, in
the sense explained by Meeks and Pérez in [9].