Maximal points with respect to cone dominance in Banach spaces and their existence

Given a convex cone in a Banach spaceV, an examination of the cone maximal points of a setX inV (with respect to the cone dominance induced by ) with respect to their characterization and existence is undertaken. The totality of cone maximal points ofX is called the conical frontier ofX. Comparisons of the conical frontiers of related sets and corresponding to related cones are made. By relaxing the compactness requirements of the underlying setX and by assuming some cone-related weaker forms of compactness, existence theorems for cone maximal points are developed. These theorems are believed to be generalizations of the existing results in one way or another.Maximizing points onX of certain linear functionals in the dual cone * of provide natural examples of cone maximal points. Properties characterizing a maximizing point of a linear functional in *, including the generalized version of Geoffrion's characterization of proper efficiency, are compiled and proved to be valid characterizations. Functionals in * with special properties are studied. Existence theorems are also obtained for the maximizing points of these functionals.The author is indebted to Professor James V. Whittaker for helpful discussions and comments and to Professors P. S. Bullen, C. W. Clark, B. N. Moyls, and F. Y. M. Wan for their encouragement and support.