A generalization of the Arrow-Barankin-Blackwell theorem in normed spaces
Let X be a normed space and C a closed convex point cone in X, \(C^+=\{x^*\in X^*:\) \(x^*(c)\geq 0\) for every \(c\in C\}\), \(C^+_ 0=\{x^*\in X^*:\) \(x^*(c)>0\) for every \(c\in C\setminus \{0\}\}\), if \(S\subset X\) and \(x^*\in C^+_ 0\) we define \(P_ S(x^*)=\{s_ 0\in S:\) \(x^*(s_ 0)\leq x^*(s)\) for every \(s\in S\}\) and \(P_ S=\cup \{P_ S(x^*):\) \(x^*\in C^+_ 0\}\), Min S\(=\{s\in S:\) \((s- C)\cap S=\{s\}\}\). If the algebraic interior of \(C^+\) is a non empty set and S is a weakly compact convex set in X, then \(P_ S\subset Min S\subset w\)-cl \(P_ S\).
scientific article; zbMATH DE number 4214799
scientific article