# Convex Functions and Spacetime Geometry

## TL;DR

This paper explores how convex functions influence the geometry of spacetimes in General Relativity, revealing restrictions on spacetime properties and opening new avenues for theoretical physics research.

## Contribution

It introduces the concept of convex functions in spacetime geometry and analyzes their implications for the structure of spacetimes in General Relativity.

## Key findings

- Convex functions impose specific geometric restrictions on spacetimes.
- Existence of convex functions affects initial data sets in General Relativity.
- The study links convexity properties to spacetime structure constraints.

## 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.