On Some Basic Results Related to Affine Functions on Riemannian Manifolds

Xiangmei Wang (), Chong Li () and Jen-Chih Yao ()
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Xiangmei Wang: Guizhou University
Chong Li: Zhejiang University
Jen-Chih Yao: China Medical University

Journal of Optimization Theory and Applications, 2016, vol. 170, issue 3, No 5, 783-803

Abstract: 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.

Keywords: Riemannian manifold; Hadamard manifold; Sectional curvature; Convex function; Quasi-convex function; Linear affine function; 52A55; 53C20; 90C25 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-016-0979-x

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