The domination property for efficiency and Bishop-Phelps theorem in locally convex spaces
Let \(X\) be a locally convex space (LCS) and \(S\) a cone in \(X\). A point \(x\) in a nonempty subset \(A\) of \(X\) is called an efficient point for \(A\) with respect to \(S\) if \((A-x)\cap (-S)=\{0\}\). The set of efficient points is denoted by \(E(S,A)\). One says that \(A\) has the domination property with respect to \(S\) if every point in \(A\) is dominated by an efficient point. For an absolutely convex subset \(B\) of \(X\), one denotes by \(E_B\) the space spanned by \(B\) equipped with the norm \(p_B\), where \(p_B\) is the Minkowski functional on \(E_B\). One says that \(B\) is a Banach disc if \((E_B,p_B)\) is a Banach space. One defines in terms of these normed subspaces localized versions of topological notions: local convergence of sequences in \(X\), local completeness of \(X\), local closedness of subsets, etc. (see, for instance, [\textit{P. Pérez Carreras} and \textit{J. Bonet}, Barrelled locally convex spaces. Amsterdam etc.: North-Holland (1987; Zbl 0614.46001)]). For a convex subset \(\Theta\) of \(X\), one also defines the notions of \(\Theta\)-closure and \(\Theta\)-closedeness, weaker than the corresponding usual notions. A \(\sigma\)-convex set (or CS-compact) is a subset \(C\) of \(X\) such that every convex series in \(C\) converges to an element of \(C\). \textit{D. H. Fremlin} and \textit{M. Talagrand} [Mathematika 26, 30--32 (1979; Zbl 0456.46011)] proved that every bounded convex \(G_\delta\)-subset of a complete LCS is \(\sigma\)-convex. In Proposition 4.2, this result is extended to locally complete LCS. The main result on the domination property (Theorem 5.1) asserts that, if \(S\) is a convex cone with a \(\sigma\)-convex base in a LCS \(X\), then every nonempty subset \(A\) of \(X\) for which there exists a linear functional \(f\) that is lower bounded on \(A\) has the domination property with respect to \(A\) (in particular, \(E(S,A)\neq\emptyset\)). Defining a notion of \(B\)-support point of a subset \(A\) of \(X\), where \(B\) is bounded convex, one proves a Bishop-Phelps type result on the density of \(B\)-support points in the local boundary of a set \(A\), provided that the set \(A\) is convex and locally closed and \(B\) is \(\sigma\)-convex.
scientific article; zbMATH DE number 6156108