
Journal of Convex Analysis 26 (2019), No. 2, 661686 Copyright Heldermann Verlag 2019 The Minimizing Vector Theorem in Symmetrized MaxPlus Algebra Cenap Özel Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia cozel@kau.edu.sa Artur Piekosz Institute of Mathematics, Cracow University of Technology, Cracow, Poland apiekosz@pk.edu.pl Eliza Wajch Inst. of Mathematics and Physics, Univ. of Natural Sciences and Humanities, Siedlce, Poland eliza.wajch@uph.edu.pl Hanifa Zekraoui Department of Mathematics, Larbi Ben M'hidi University, Oum El Bouaghi, Algeria hzekraoui421@gmail.com Assuming ZF and its consistency, we study some topological and geometrical properties of the symmetrized maxplus algebra in the absence of the axiom of choice in order to discuss the minimizing vector theorem for finite products of copies of the symmetrized maxplus algebra. Several relevant statements that follow from the axiom of countable choice restricted to sequences of subsets of the real line are shown. Among them, it is proved that if all simultaneously complete and connected subspaces of the plane are closed, then the real line is sequential. A brief discussion about semidenrites is included. Older known proofs in ZFC of several basic facts relevant to proximinal and Chebyshev sets in metric spaces are replaced by new proofs in ZF. It is proved that a nonempty subset C of the symmetrized maxplus algebra is Chebyshev in this algebra if and only if C is simultaneously closed and connected. An application of it to a version of the minimizing vector theorem for finite products of the symmetrized maxplus algebra is shown. Open problems concerning some statements independent of ZF and other statements relevant to Chebyshev sets are posed. Keywords: Symmetrized maxplus algebra, metric, complete metric, Cantor complete metric, proximinal set, Chebyshev set, convexity, geometric convexity, minimizing vector theorem, semidendrite, ZF, axiom of countable choice for the real line, independence results. MSC: 15A80, 16Y60; 03E25, 54F15 [ Fulltextpdf (191 KB)] for subscribers only. 