Convex Functions and Spacetime Geometry

Abstract

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.

In the light of recent renewed interest in this topic (see arxiv: 1510.04228 and 1702.02652) we are submitting this brief summary of our old results presented by A.I. at the International Workshop: Frontiers of Cosmology and Gravitation held in Kyoto in 2001.and published in the proceedings. See also arXiv:gr-qc/0011055