On the continuity of geodesically convex functions on Riemannian manifolds
Differential Geometry
2026-01-06 v2
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.
Cite
@article{arxiv.2512.05621,
title = {On the continuity of geodesically convex functions on Riemannian manifolds},
author = {Victor-Emmanuel Brunel and Pierre Pansu},
journal= {arXiv preprint arXiv:2512.05621},
year = {2026}
}