Infinite-dimensional convex cones: internal geometric structure and analytical representation

The main purpose of this paper is to study an internal geometric structure of convex cones in infinite-dimensional vector spaces and to obtain an analytical description of those. We suppose that a convex cone has an elementary (one-piece) internal geometric structure if it is relatively algebraically open. It is proved that an arbitrary convex cone is the disjoint union of the partial ordered family of its relatively algebraically open convex subcones and, moreover, as an ordered set this family is an upper semilattice. We identify the structure of this upper semilattice with the internal geometric structure of the corresponding convex cone. Theorem 16 and Example 17 demonstrate that the internal geometric structure of a convex cone is related to its facial structure but in an infinite-dimensional setting these two structures may differ each other. Further, we study the internal geometric structure of conical halfspaces (convex cones whose complements also are convex cones). We show that every conical halfspace is the disjoint union of the linearly ordered family of relatively algebraically open convex subcones each of which is a conical halfspace in its linear hull. Using the internal geometric structure of conical halfspaces, each conical halfspace is associated with a linearly ordered family of linear functions, which generates in turn a real-valued function, called a step-linear one, analytically describing this conical halfspace. At last, we establish that an arbitrary asymmetric convex cone admits an analytical representation by the family of step-linear functions.