The minimizing vector theorem in symmetrized max-plus algebra / Cenap Özel, Artur Piękosz, Eliza Wajch, Hanifa Zekraoui // Journal of Convex Analysis [Dokument elektroniczny]. – 2019, Vol. 26, No. ... więcej

The minimizing vector theorem in symmetrized max-plus algebra / Cenap Özel, Artur Piękosz, Eliza Wajch, Hanifa Zekraoui // Journal of Convex Analysis [Dokument elektroniczny]. – 2019, Vol. 26, No. 2, s. 661-686. – Tryb dostępu: http://www.heldermann-verlag.de/jca/jca26/jca1872-b.pdf. – Bibliogr. 23 poz., Streszcz. ang. – ISSN 2363-6394

Assuming ZF and its consistency, we study some topological and geometrical properties of thesymmetrized max-plus algebra in the absence of the axiom of choice in order to discuss theminimizing vector theorem ... więcej

Assuming ZF and its consistency, we study some topological and geometrical properties of thesymmetrized max-plus algebra in the absence of the axiom of choice in order to discuss theminimizing 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 sequencesof 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. Abrief discussion about semidenrites is included. Older known proofs in ZFC of several basicfacts 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 Chebyshevin 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.