What Do ‘Convexities’ Imply on Hadamard Manifolds?

Alexandru Kristály (), Chong Li (), Genaro López-Acedo () and Adriana Nicolae ()
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Alexandru Kristály: Babeş-Bolyai University
Chong Li: Zhejiang University
Genaro López-Acedo: Universidad de Sevilla
Adriana Nicolae: Babeş-Bolyai University

Journal of Optimization Theory and Applications, 2016, vol. 170, issue 3, No 21, 1068-1074

Abstract: 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.

Keywords: Hadamard manifold; Convexity; Affin function; Convex hull; 53C23; 53C24 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10957-015-0780-2

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