  In many applications it is important to establish if a given topological
preordered space has a topology and a preorder which can be recovered from the
set of continuous isotone functions. Under antisymmetry this property, also
known as quasi-uniformizability, allows one to compactify the topological space
and to extend its order dynamics. In this work we study locally compact
$\sigma$-compact spaces endowed with a closed preorder. They are known to be
normally preordered, and it is proved here that if they are locally convex,
then they are convex, in the sense that the upper and lower topologies generate
the topology. As a consequence, under local convexity they are
quasi-uniformizable. The problem of establishing local convexity under
antisymmetry is studied. It is proved that local convexity holds provided the
convex hull of any compact set is compact. Furthermore, it is proved that local
convexity holds whenever the preorder is compactly generated, a case which
includes most examples of interest, including preorders determined by cone
structures over differentiable manifolds. The work ends with some results on
the problem of quasi-pseudo-metrizability. As an application, it is shown that
every stably causal spacetime is quasi-uniformizable and every globally
hyperbolic spacetime is strictly quasi-pseudo-metrizable.
