  We show that in Lorentzian manifolds, sectional curvature bounds of the form
$\mathcal{R}\le K\,$, as defined by Andersson and Howard, are closely tied to
space-time convex and $\lambda$-convex ($\lambda>0$) functions, as defined by
Gibbons and Ishibashi. Among the consequences are a natural construction of
such functions, and an analogue, that applies to domains of a new type, of a
theorem of Al\'ias, Bessa and deLira ruling out trapped submanifolds.
