Concavifying the Quasiconcave

We revisit a classic question of Fenchel from 1953: Which quasiconcave functions can be concavified by post-composition with a monotonic transformation? This question has a long history in economic utility theory. While many authors have given partial answers under various assumptions, we offer a complete characterization for quasiconcave functions without a priori assumptions on regularity. The answer hinges on the local regularity class of the function. We establish this characterization of concavifiability for continuous functions whose domain is any arbitrary geodesic metric space. Under the additional assumption of twice differentiability, we also provide simpler necessary and sufficient conditions for concavifiability on Riemannian manifolds which essentially generalize those given by Kannai for the Euclidean case.