  This paper explores the relation between convex functions and the geometry of
space-times and semi-Riemannian manifolds (an investigation initiated by
Gibbons-Ishibashi). Specifically, we study geodesic connectedness. We give
geometric-topological proofs of geodesic connectedness for classes of
space-times to which known methods do not apply. For instance: A
null-disprisoning space-time is geodesically connected if it supports a proper,
nonnegative strictly convex function whose critical set is a point. Timelike
strictly convex hypersurfaces of Minkowski space are geodesically connected. We
also give a criterion for the existence of a convex function on a
semi-Riemannian manifold. We compare our work with previously known results.
