Non-existence of concave functions on certain metric spaces

TL;DR

This paper extends classical results about the non-existence of non-trivial concave functions from complete manifolds to various metric spaces, including Alexandrov spaces and Hölder Riemannian manifolds.

Contribution

It proves analogous non-existence theorems for concave functions on several classes of metric spaces, broadening the scope of Yau's original result.

Findings

No non-trivial continuous concave functions exist on Alexandrov spaces with curvature bounds.

Analogous non-existence results hold for Hölder Riemannian manifolds.

Theorems extend classical manifold results to more general 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, $C^{\alpha}$-H\"older Riemannian manifolds.