Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Cenap Özel

Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

cozel@kau.edu.sa

Artur Piękosz

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

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

Mathematics Subject Classification

15A80, 16Y60; 03E25, 54F15

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Suggested citation

C. Özel, A. Piękosz, E. Wajch, H. Zekraoui. “The Minimizing Vector Theorem in Symmetrized Max-Plus Algebra.” Journal of Convex Analysis 26 (2019), No. 2, 661–686. https://doi.org/10.68381/jca26035

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.