On the continuity of geodesically convex functions on Riemannian manifolds
Victor-Emmanuel Brunel, Pierre Pansu
TL;DR
This paper addresses the regularity of geodesically convex functions on Riemannian manifolds, proving they are continuous (indeed locally Lipschitz) in the interior of their domain. To overcome gaps in earlier proofs, the authors develop a complete argument based on iterated barycenters and iterated barycentric hulls within totally normal neighborhoods, and use an implicit function theorem to show a neighborhood lies inside an iterated simplex, yielding local upper bounds. Consequently, geodesically convex functions are uniformly bounded above on small balls, which via standard convex-analysis arguments gives continuity. The paper also discusses the extent of these results beyond Riemannian spaces, presenting counterexamples in general geodesic spaces and introducing the geodesic interior concept, thus clarifying the limits of regularity in non-Riemannian settings.
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.