Journal of Convex Analysis 29 (2022), No. 2, 519--529
Copyright Heldermann Verlag 2022
Hyperplane Separation Type Theorems in Hadamard Manifolds
Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
Fakultät für Mathematik, Universität Duisburg-Essen, Essen, Germany
We prove a hyperplane separation theorem for disjoint and convex sets in two-dimensional Hadamard manifolds. Furthermore, in higher dimensions we prove such a theorem if additionally the two involved sets are closed. This extends a recent generalization of J. Shenawy [Horosphere slab separation theorems in manifolds without conjugate points, J. Egyptian Math. Soc. 27 (2019)] and R. Bergmann et al. [Fenchel duality and a separation theorem on Hadamard manifolds, SIAM J. Optimization (2022)] of the hyperplane separation theorem in Euclidean space to Hadamard manifolds at which one of the disjoint sets is assumed to be convex and compact and the other convex and closed; the proof relies on results of these references. Note that generalizations of the hyperplane separation theorem from Euclidean space to topological vector spaces are the classical and well-known Hahn-Banach separation theorems.
Keywords: Hadamard manifolds, separation theorem.
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