On the continuity of geodesically convex functions on Riemannian manifolds

TL;DR

This paper proves that geodesically convex functions on Riemannian manifolds are continuous within their domain's interior, addressing a gap in the existing proof and exploring potential extensions beyond Riemannian geometry.

Contribution

The authors provide a complete proof of the continuity of geodesically convex functions and discuss extensions beyond Riemannian manifolds.

Findings

All geodesically convex functions are continuous in the interior of their domain.

Identified and filled a gap in the existing proof of this continuity.

Discussed possible extensions of the result beyond Riemannian manifolds.

Abstract

In this short note, we prove that all geodesically convex functions defined on a Riemannian manifold are continuous in the interior of their domain. This is a folklore result, but to the best of our knowledge, there is only one available proof, which is largely cited. However, it contains a significant gap, which we fill here. We also discuss extensions of this result beyond the Riemannian setting.