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Global Busemann concavity of Berwald spaces with lower flag curvature bound

Determine whether every Berwald space with flag curvature bounded below satisfies Busemann concavity globally, not just locally; i.e., establish or refute that Busemann concavity holds for all geodesics and all scales in such Berwald spaces.

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Background

The paper discusses the relationship between synthetic curvature conditions and Finsler structures. It notes that Berwald spaces with lower flag curvature bounds satisfy Busemann concavity locally, but a global extension is not established.

Resolving global Busemann concavity for Berwald spaces would bridge local and global synthetic curvature behaviors in Finsler geometry and clarify the scope of Busemann convexity/concavity frameworks across important subclasses.

References

While one can show that Berwald spaces with flag curvature bounded below satisfy Busemann concavity locally, it remains unknown whether all such Berwald spaces satisfy Busemann concavity globally, as pointed out by Kell Section 2.1.

— On the Structure of Busemann Spaces with Non-Negative Curvature (2508.12348 - Han et al., 17 Aug 2025) in Section 3 (S-concavity, local semi-convexity and Busemann concavity), Remark (Berwald spaces)