On the continuity of geodesically convex functions on Riemannian manifolds

Victor-Emmanuel Brunel; Pierre Pansu

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arXiv · 2512.05621

On the continuity of geodesically convex functions on Riemannian manifolds

Victor-Emmanuel Brunel·Pierre Pansu

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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.

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BibTeX RIS

Victor-Emmanuel Brunel, Pierre Pansu. 2025-12-05. On the continuity of geodesically convex functions on Riemannian manifolds. https://arxiv.org/abs/2512.05621

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