Concavifying the Quasi-Concave

This is a 1/3 length math version of a previous paper with the same title. We revisit a classic question of Fenchel from 1953: Which quasiconcave functions can be concavified by a monotonic transformation? While many authors have given partial answers under various assumptions, we offer a complete characterization for all quasiconcave functions without a priori assumptions on regularity. In particular, we show that if and only if a real-valued function f is strictly quasiconcave, (except possibly for a flat interval at its maximum) and furthermore belongs to a certain explicitly determined regularity class, there exists a strictly monotonically increasing functiong such that g\smallcircle f is strictly concave. Our primary new contribution is determining this precise minimum regularity class.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. Under the assumption of twice differentiability, we also establish simpler sufficient conditions for concavifiability on arbitrary Riemannian manifolds, which essentially generalizes those given by Kannai in 1977 for the Euclidean case. Lastly we present the approximation result that if a function f is either weakly or strongly quasiconcave then there exists an arbitrarily close strictly concavifiable approximationh to f