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

A subset

of

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

is B

^{-1} − 1

-convex if for all

x_{1},x_{2}\in A x 1 , x 2 ∈ A

and all

t\geq1 t ≥ 1

one has

tx_{1}\wedge x_{2}\in A t x 1 ∧ x 2 ∈ A

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

^{-1}- − 1 −

convex sets and B

^{-1}- − 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} − 1

-convex sets.