Convex functions and spacetime geometry

Convexity and convex functions play an important role in theoretical physic s. 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 f unction. We show how the existence of a convex function on a spacetime plac es restrictions on the properties of the spacetime geometry.