Journal of Convex Analysis 26 (2019), No. 2, 661--686
Copyright Heldermann Verlag 2019
The Minimizing Vector Theorem in Symmetrized Max-Plus 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 max-plus 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 max-plus 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 max-plus 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 max-plus algebra is shown. Open problems concerning some statements independent of ZF and other statements relevant to Chebyshev sets are posed.
Keywords: Symmetrized max-plus 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
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