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Journal of Convex Analysis 21 (2014), No. 2, 571--580
Copyright Heldermann Verlag 2014

Separation of B-1-Convex Sets by B-1-Measurable Maps

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

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

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


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].\par In this paper, we establish separation and a Hahn-Banach-like Theorem for B$^{-1}$-convex sets.

Keywords: B-convexity, half spaces, gauges, co-gauges, separation, B-measurable maps.

MSC: 52A30, 52A01, 52A41, 26B25

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