On the continuity of geodesically convex functions on Riemannian manifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908748151259136 |
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| author | Brunel, Victor-Emmanuel Pansu, Pierre |
| author_facet | Brunel, Victor-Emmanuel Pansu, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the continuity of geodesically convex functions on Riemannian manifolds Brunel, Victor-Emmanuel Pansu, Pierre Differential Geometry 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. |
| title | On the continuity of geodesically convex functions on Riemannian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.05621 |