Non-existence of concave functions on certain metric spaces

Abstract

In the paper \cite{yau1974convex}, Yau 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, $C^{\alpha }$-Hölder Riemannian manifolds.

19pages