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

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.