On the continuity of geodesically convex functions on Riemannian manifolds

2025/12/05 by Brunel, Victor-Emmanuel, Pansu, Pierre
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.05621

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.

Citations

• Basic convex analysis in metric spaces with bounded curvature

Related

• Convexity and some geometric properties
• Geodesic Sandwich Theorem with an Application
• Convexity of the distance function to convex subsets of Riemannian manifolds
• Convex Functions on Sub-Riemannian Manifolds. I
• Geodesic Convexity Types in Riemannian Manifolds