  Convexity and convex functions play an important role in theoretical physics.
To initiate a study of the possible uses of convex functions in General
Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an
initial data set $(\Sigma, h_{ij}, K_{ij})$ admitting a suitably defined convex
function. We show how the existence of a convex function on a spacetime places
restrictions on the properties of the spacetime geometry.
