Two generalizations of a theorem of Arrow, Barankin, and Blackwell

摘要: In 1953, Arrow, Barankin, and Blackwell proved that if $R^n $ is equipped with its natural ordering A a closed convex subset of $, then the set points in can be supported by strictly positive linear functionals dense all efficient (maximal) A. this note two generalizations result are given. The first these setting dual system requires relatively weak assumptions on cone but rather strong compactness assumption second generalization, which locally space, relaxes demands more stringent cone. This was recently obtained Petschke for normed spaces [M. Petschke, “On theorem Barankin Blackwell”, SIAM J. Control Optim., 28 (1990), pp. 395–401]. proof given here substantially different from Petschke.