Abstract
The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.
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Acknowledgements
The authors would like to express their gratitude to referee for valuable suggestions on the improvement of the paper. João X. Cruz Neto and Paulo A. Sousa were partially supported by CNPq/Brazil Grant Nos. 305462/2014-8 and 304823/2013-9.
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Communicated by Sándor Zoltán Németh.
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Neto, J.X.d.C., Melo, Í.D.L. & Sousa, P.A.A. Convexity and Some Geometric Properties. J Optim Theory Appl 173, 459–470 (2017). https://doi.org/10.1007/s10957-017-1087-2
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DOI: https://doi.org/10.1007/s10957-017-1087-2