Some properties of cones in normed spaces and their application to investigating extremal problems

Abstract

In the present paper, the notions of cones of the same sense and cones of the opposite sense are introduced. The basic properties of these cones are examined. The most essential property is the following: ifX is a Banach space, then, in the dual spaceX*, the algebraic sum of a finite number of weakly * closed convex cones of the same sense is weakly * closed. On the basis of the properties of these cones, some necessary conditions for the existence of extrema in extremal problems are given.