Abstract

B\mathbb{B}-convexity was defined by the author and C. D. Horvath [B\mathbb{B}-convexity, Optimization 53(2) (2004) 103–127] as a suitable Painlevé-Kuratowski limit of linear convexities. Recently, an alternative algebraic formulation over the whole Euclidean vector space was proposed by the author in further articles [Some remarks on an idempotent and non-associative convex structure, J. Convex Analysis 22 (2015) 259–289; and Separation properties in some idempotent and symmetrical convex structure, J. Convex Analysis 24 (2017) 1143–1168]. In this paper the structure of the polytopes arising in such a context is analyzed. It is first established that this notion of convexity has the interval and decompositions properties but fails the recursive property. Among the key results of the paper, it is proven that a polytope defined over Rn\mathbb{R}^n is the Painlevé-Kuratowski limit of a suitable sequence of generalized convex polytopes. Some analogues to the Helly, Radon and Caratheodory theorems are derived in the limit. A separation theorem for B\mathbb{B}-polytopes is stated extending an earlier result recently established by the author [Separation properties in some idempotent and symmetrical convex structure, J. Convex Analysis 24 (2017) 1143–1168] for fixed B\mathbb{B}-convex sets. Finally, considering the Kuratowski-Painlevé limit of a suitable sequence of linear halfspaces the external representation of a B\mathbb{B}-polytope is established over Rn\mathbb{R}^n. It follows that the interior of a B\mathbb{B}-polytope defined over Rn\mathbb{R}^n satisfies the convexity criterion proposed by the author previously [Some remarks on an idempotent and non-associative convex structure, J. Convex Analysis 22 (2015) 259–289].

Author information

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

Walter Briec

LAMPS-Laboratory of Mathematics and Physics, University of Perpignan, 52 avenue Villeneuve, 66000 Perpignan, France

briec@univ-perp.fr

Keywords

  • Generalized mean
  • convexity
  • convex hull
  • duality
  • semilattice
  • B-convexity

Mathematics Subject Classification

06D50, 32F17

References

  1. G. Adilov, A. M. Rubinov: B-convex sets and functions, Numer. Functional Analysis Optim. 27 (2006) 237–257.https://doi.org/10.1080/01630560600698343Google Scholar
  2. G. Adilov, I. Yesilce: B−1 -convex functions, J. Convex Analysis 24(2) (2017) 505–517.https://doi.org/10.68381/jca24033Google Scholar
  3. M. Avriel: R-convex functions, Math. Program. 2(1) (1972) 309–323.https://doi.org/10.1007/BF01584551Google Scholar
  4. M. Avriel: Nonlinear Programming: Analysis and Methods, Prentice Hall, Englewood Cliffs (1976).Google Scholar
  5. A. Ben-Tal: On generalized means and generalized convex functions, J. Optim. Theory Appl. 21(1) (1977) 1–13.https://doi.org/10.1007/BF00932539Google Scholar
  6. W. Briec: Some remarks on an idempotent and non-associative convex structure, J. Convex Analysis 22 (2015) 259–289.https://doi.org/10.68381/jca22013Google Scholar
  7. W. Briec: Separation properties in some idempotent and symmetrical convex structure, J. Convex Analysis 24(4) (2017) 1143–1168.https://doi.org/10.68381/jca24067Google Scholar
  8. W. Briec: Some remarks on the notion of determinant in an idempotent, symmetrical and non-associative algebraic structure, working paper.Google Scholar
  9. W. Briec, C. D. Horvath: B-convexity, Optimization 53(2) (2004) 103–127.https://doi.org/10.1080/02331930410001695283Google Scholar
  10. W. Briec, C. D. Horvath: Nash points, Ky Fan inequality and equilibria of abstract economies in max-plus and B-convexity, J. Math. Anal. Appl. 341 (2008) 188–199.https://doi.org/10.1016/j.jmaa.2007.09.056Google Scholar
  11. W. Briec, C. D. Horvath: On the separation of convex sets in some idempotent semimodules, Linear Algebra Appl. 435 (2011) 1542–1548.https://doi.org/10.1016/j.laa.2010.01.018Google Scholar
  12. W. Briec, C. D. Horvath, A. Rubinov: Separation in B-convexity, Pacific J. Optimization 1 (2005) 13–30.Google Scholar
  13. G. H. Hardy, J. E. Littlewood, G. Pólya: Inequalities, Cambridge University Press, Cambridge (1952).Google Scholar
  14. S. Kemali, I. Yesilce, G. Adilov: B -convexity, B−1 -convexity, and their comparison, Numer. Functional Analysis Optim. 36(2) (2015) 133–146.https://doi.org/10.1080/01630563.2014.970641Google Scholar
  15. V. P. Maslov, S. N. Samborski (eds.): Idempotent Analysis, Advances in Soviet Mathematics, American Mathematical Society, Providence, (1992).https://doi.org/10.1090/advsov/013Google Scholar
  16. M. Patriche: Fixed point and equilibrium theorems in a generalized convexity framework, Journal Optim. Theory Appl. 156 (2013) 701–715.https://doi.org/10.1007/s10957-012-0133-3Google Scholar
  17. A. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht et al. (2000).https://doi.org/10.1007/978-1-4757-3200-9Google Scholar
  18. G. Tinaztepe, I. Yesilce, G. Adilov: Separation of B−1 -convex sets by B−1 - measurable maps, J. Convex Analysis 21(2) (2014) 571–580.https://doi.org/10.68381/jca21030Google Scholar
  19. M. Van de Vel: Theory of Convex Structures, North Holland Mathematical Library 50, Elsevier, Amsterdam et al. (1993).Google Scholar

Suggested citation

W. Briec. “On some Class of Polytopes in an Idempotent, Symmetrical and Non-Associative Convex Structure.” Journal of Convex Analysis 26 (2019), No. 3, 823–853. https://doi.org/10.68381/jca26044

Download BibTeXPublished by Heldermann Verlag.

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