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 an explicitly determined regularity class, does there exist a strictly monotonically increasing function g such that g • 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 • 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 concavifiability on Euclidean spaces and other Riemannian manifolds, which suffice for most applications.
In this paper we explore the asymptotic geometry of negatively curved homogeneous manifolds. We characterize asymptotic harmonicity in terms of various natural measures. We also show that the Bowen-Margulis, harmonic, and Liouville measures on the unit tangent bundle and the corresponding measures on the boundary are always in the same measure class. We then show that Cheeger's constant, the Kaimanovich entropy, and the bottom of the spectrum are all maximal for these spaces. Along the way we present sharp asymptotic estimates for Jacobi fields and the Poisson and Green's kernels. Finally, we present examples showing that in general these manifolds are not asymptotically harmonic.
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