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What Do ‘Convexities’ Imply on Hadamard Manifolds?

Author

Listed:
  • Alexandru Kristály

    (Babeş-Bolyai University
    Óbuda University)

  • Chong Li

    (Zhejiang University)

  • Genaro López-Acedo

    (Universidad de Sevilla)

  • Adriana Nicolae

    (Babeş-Bolyai University
    Simion Stoilow Institute of Mathematics of the Romanian Academy)

Abstract

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper, we prove that these conditions are mutually equivalent and they hold, if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well-known Euclidean counterparts.

Suggested Citation

  • Alexandru Kristály & Chong Li & Genaro López-Acedo & Adriana Nicolae, 2016. "What Do ‘Convexities’ Imply on Hadamard Manifolds?," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1068-1074, September.
    • Handle: RePEc:spr:joptap:v:170:y:2016:i:3:d:10.1007_s10957-015-0780-2
    DOI: 10.1007/s10957-015-0780-2
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    References listed on IDEAS

    as
    1. Li-Wen Zhou & Nan-Jing Huang, 2013. "Existence of Solutions for Vector Optimization on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 44-53, April.
    2. E. Papa Quiroz, 2013. "An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 56(1), pages 43-59, May.
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