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On Some Basic Results Related to Affine Functions on Riemannian Manifolds

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Abstract

We study some basic properties related to affine functions on Riemannian manifolds. A characterization for a function to be linear affine is given and a counterexample on Poincaré plane is provided, which, in particular, shows that assertions (i) and (ii) claimed by Papa Quiroz in (J Convex Anal 16(1):49–69, 2009, Proposition 3.4) are not true, and that the function involved in assertion (ii) is indeed not quasi-convex. Furthermore, we discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures.

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Acknowledgments

The authors are grateful to both anonymous reviewers for their valuable suggestions and remarks. Research of the first author is supported in part by the Scientific Research Projects of Guizhou University (Grant 201406). Research of the second author is supported in part by the National Natural Science Foundation of China (Grants 11571308, 11371325). Research of the third author is supported in part by the Grant MOST 102-2115-M-039-003-MY3.

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Authors and Affiliations

  1. College of Science, Guizhou University, Guiyang, 550025, People’s Republic of China

    Xiangmei Wang

  2. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Chong Li

  3. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

Authors
  1. Xiangmei Wang
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  2. Chong Li
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  3. Jen-Chih Yao
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Correspondence to Chong Li.

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Communicated by Sándor Zoltán Németh.

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Wang, X., Li, C. & Yao, JC. On Some Basic Results Related to Affine Functions on Riemannian Manifolds. J Optim Theory Appl 170, 783–803 (2016). https://doi.org/10.1007/s10957-016-0979-x

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  • DOI: https://doi.org/10.1007/s10957-016-0979-x

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