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edited Sep 13, 2015 at 14:46

For complete Riemannian manifolds of bounded sectional curvature the answer is YES by L.F. Tam. Exhaustion functions on complete manifolds MR2648946. Then the answer for the Hilbert metric on a properly convex manifold is also YES since such a manifold admits a Riemannian metric of bounded curvature that is bi-Lipshitz to the Hilbert metric

answered Sep 13, 2015 at 14:31

Daryl Cooper