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Convexity and Some Geometric Properties

Author

Listed:
  • João Xavier da Cruz Neto

    (Universidade Federal do Piauí)

  • Ítalo Dowell Lira Melo

    (Universidade Federal do Piauí)

  • Paulo Alexandre Araújo Sousa

    (Universidade Federal do Piauí)

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.

Suggested Citation

  • João Xavier da Cruz Neto & Ítalo Dowell Lira Melo & Paulo Alexandre Araújo Sousa, 2017. "Convexity and Some Geometric Properties," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 459-470, May.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-017-1087-2
    DOI: 10.1007/s10957-017-1087-2
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    References listed on IDEAS

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    1. X. M. Wang & C. Li & J. C. Yao, 2015. "Subgradient Projection Algorithms for Convex Feasibility on Riemannian Manifolds with Lower Bounded Curvatures," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 202-217, January.
    2. O. P. Ferreira & P. R. Oliveira, 1998. "Subgradient Algorithm on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 93-104, April.
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