Concavifying the QuasiConcave

We show that if and only if a real-valued function f is strictly quasiconcave except possibly for a at interval at its maximum, and furthermore belongs to an explicitly determined regularity class, does there exist a strictly monotonically increasing function g such that g o f is strictly concave. Moreover, if and only if the function f is either weakly or strongly quasiconcave there exists an arbitrarily close approximation h to f and a monotonically increasing function g such that g o h is strictly concave. We prove this sharp characterization of quasiconcavity for continuous but possibly nondifferentiable functions whose domain is any Euclidean space or even any arbitrary geodesic metric space. While the necessity that f belong to the special regularity class is the most surprising and subtle feature of our results, it can also be difficult to verify. Therefore, we also establish a simpler sufficient condition for concaviability on Euclidean spaces and other Riemannian manifolds, which suffice for most applications.