Separation of B^{-1}-Convex Sets by B^{-1}-Measurable Maps

Gultekin Tinaztepe; Ilknur Yesilce; Gabil Adilov

doi:10.68381/jca21030

Abstract

A subset

A

of

\mathbb{R}^{n}_{++}

is B

^{-1}

-convex if for all

x_{1},x_{2}\in A

and all

t\geq1

one has

tx_{1}\wedge x_{2}\in A

. These sets were first investigated in papers of G. Adilov and I. Yesilce [``B

^{-1}-

convex sets and B

^{-1}-

measurable maps'', Numerical Functional Analysis and Optimization 33(2) (2012) 131–141; ``On Generalization of the Concept of Convexity'', Hacettepe Journal of Mathematics and Statistics 41(5) (2012) 723–730], and of W. Briec and Q. B. Liang [``On Some Semilattice Structures for Production Technologies'', European Journal of Operational Research 215 (2011) 740–749]. In this paper, we establish separation and a Hahn-Banach-like Theorem for B

^{-1}

-convex sets.

Author information

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

Gultekin Tinaztepe

Vocational School of Technical Sciences, Akdeniz University, Dumlupinar Boulevard, 07058 Campus Antalya, Turkey

gtinaztepe@akdeniz.edu.tr

Ilknur Yesilce

Faculty of Science and Letters, Mersin University, Ciftlikkoy Campus, 33343 Mersin, Turkey

ilknuryesilce@gmail.com

Gabil Adilov

Faculty of Education, Akdeniz University, Dumlupinar Boulevard, 07058 Campus Antalya, Turkey

gabiladilov@gmail.com

Keywords

B-convexity
half spaces
gauges
co-gauges
separation
B-measurable maps

Mathematics Subject Classification

52A30, 52A01, 52A41, 26B25

References

G. Adilov, A. Rubinov: B-convex sets and functions, Numer. Funct. Anal. Optim. 27(3–4) (2006) 237–257.https://doi.org/10.1080/01630560600698343 Google Scholar
G. Adilov, I. Yesilce: B−1 -convex sets and B−1 -measurable maps, Numer. Funct. Anal. Optim. 33(2) (2012) 131–141.https://doi.org/10.1080/01630563.2011.618960 Google Scholar
G. Adilov, I. Yesilce: On generalizations of the concept of convexity, Hacet. J. Math. Stat. 41(5) (2012) 723–730.Google Scholar
W. Briec, C. D. Horvath: B-convexity, Optimization 53 (2004) 103–127.https://doi.org/10.1080/02331930410001695283 Google Scholar
W. Briec, C. D. Horvath: Halfspaces and Hahn-Banach like properties in B-convexity and max-plus convexity, Pac. J. Optim. 4 (2008) 293–317.Google Scholar
W. Briec, C. D. Horvath, A. M. Rubinov: Separation in B-convexity, Pac. J. Optim. 1 (2005) 13–30.Google Scholar
W. Briec, Q. B. Liang: On some semilattice structures for production technologies, Eur. J. Oper. Res. 215 (2011) 740–749.https://doi.org/10.1016/j.ejor.2011.06.012 Google Scholar
A. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2014. Rights now held by Banach Press.

[1] G. Adilov, A. Rubinov: B-convex sets and functions, Numer. Funct. Anal. Optim. 27(3–4) (2006) 237–257.https://doi.org/10.1080/01630560600698343 Google Scholar

[2] G. Adilov, I. Yesilce: B−1 -convex sets and B−1 -measurable maps, Numer. Funct. Anal. Optim. 33(2) (2012) 131–141.https://doi.org/10.1080/01630563.2011.618960 Google Scholar

[3] G. Adilov, I. Yesilce: On generalizations of the concept of convexity, Hacet. J. Math. Stat. 41(5) (2012) 723–730.Google Scholar

[4] W. Briec, C. D. Horvath: B-convexity, Optimization 53 (2004) 103–127.https://doi.org/10.1080/02331930410001695283 Google Scholar

[5] W. Briec, C. D. Horvath: Halfspaces and Hahn-Banach like properties in B-convexity and max-plus convexity, Pac. J. Optim. 4 (2008) 293–317.Google Scholar

[6] W. Briec, C. D. Horvath, A. M. Rubinov: Separation in B-convexity, Pac. J. Optim. 1 (2005) 13–30.Google Scholar

[7] W. Briec, Q. B. Liang: On some semilattice structures for production technologies, Eur. J. Oper. Res. 215 (2011) 740–749.https://doi.org/10.1016/j.ejor.2011.06.012 Google Scholar

[8] A. Rubinov: Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000).https://doi.org/10.1007/978-1-4757-3200-9 Google Scholar