Properties And Application Of Agmnontod Dual Cones In A Locally Convex Space

In this paper, we discuss properies and application of augmented cone in locallyconvex spaces. Firstly, by extending a usual definition of dual cone in locally convexspaces, augmented dual cones are introduced. A special class of monotonicallyincreasing sublinear functions is defined by using the elements of the augmented dualcone. Let C be a closed convex cone of locally convex spaces, if C has a boundedbase, then cone(-C) can be represented as a zero level set of the monotonicallyincreasing sublinear functions. As the same way, we show the same conclusionusing diferent methods in Banach spaces. Secondly, we show without convexityassumption that two close cones possessing a certain separation property cab beseperated by using a zero sublevel set of some function from this class. Under certainconditions, the opposite conclusion also holds. It is shown also that any closed conepossessing the separation property together with its ε-conic neighborhood can beapproximated arbitrarily closely by a zero sublevel set of some function from thisclass. As an application, a simple and efcient scalarization technique is shown thatany properly minimal point of a set in local convex spaces can be calculated byminiminizing a certain sublinear functional.