Non-existence of concave functions on certain metric spaces

Abstract

Yau [Math. Ann. 207 (1974), pp. 269–270] proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces with curvature bounded below/above, C α C^{\alpha } -Hölder Riemannian manifolds.