QUASI-INTERIOR POINTS OF CONES

Abstract

Abstract : A point is not equal to theta belonging to a convex cone K with vertex theta in a locally convex linear topological space X is called a quasi interior point (QI-point) of K if the linear extension of the set K intersection (x-K) is dense in X. The set K sub q of all quasi-interior points of K is called the quasi-interior of K. Many properties of QI-points and of cones with non-void quasi-interiors are determined. Among the results established are the following. If K has a non-void interior K(o) then K sub q = K(o). Examples are given to show that a cone with void interior may have a non- void quasi-interior. Let K and K' be cones with non-void quasi-interiors K sub q and K' sub q such that K sub q intersection K' sub q = null set. If H is a hyperplane separating K and K' then H strictly separates K sub q and K'sub q. Each QI-point of K is a non-support point of K. If K sub q is not equal to null set and C is a convex set with non-void interior C(o) such that C(o) intersection K sub q = null set then there exists a hyperplane H separating C and K and strictly separating C(o) and K sub q.