On the locally biorder-convex space and super efficiency
A topological vector space is said to be ordered if a convex cone of positive elements is specified. Standard references for such spaces are the books by \textit{H. H. Schaefer} [``Topological vector spaces'', Macmillan (1966; Zbl 0141.30503)] and \textit{G. Jameson} [``Ordered linear spaces'', Lect. Notes Math. 141 (1970; Zbl 0196.13401)]. Key notions in the subject are order intervals, i.e. \([a,b]:=\{x:a\leq x\leq b\}=(a+K)\cap(b-K)\), normal cones and the decomposition property. A basic theorem connects the normality of \(K\) with the decomposition property for the dual space. In this paper the author considers a bi-ordered space for which two convex cones, \(C_1\) and \(C_2\) are specified. These are used to generalize the notion of order interval to: \[ [a,b]^{C_1}_{C_2}:=\{x:a\leq_{C_1}x\leq_{C_2}b\}=(a+C_1)\cap(b-C_2). \] This leads to a generalization of normality as an interesting relationship between \(C_1\) and \(C_2\), and a generalization of the theorem mentioned above. The paper is concisely written with the interplay between geometry and analysis clearly presented.
scientific article; zbMATH DE number 1283805