Infinite-dimensional Convex Cones: Internal Geometric Structure and Analytical Representation

TL;DR

This paper explores the internal geometric structure of convex cones in infinite-dimensional spaces without topology, revealing their decomposition into open components and providing an analytical representation via step-linear functions.

Contribution

It introduces the concept of open components, characterizes the internal geometric structure, and establishes an analytical representation for asymmetric convex cones in infinite dimensions.

Findings

Convex cones decompose into disjoint open components forming an upper semilattice.

Internal geometric structure relates to but can differ from facial structure in infinite dimensions.

Asymmetric convex cones can be represented analytically using step-linear functions.

Abstract

In the paper we consider convex cones in infinite-dimensional real vector spaces which are endowed with no topology. The main purpose is to study an internal geometric structure of convex cones and to obtain an analytical description of those. To this end, we first introduce the notion of an open component of a convex cone and then prove that an arbitrary convex cone is the disjoint union of the partial ordered family of its open components 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 a convex cone. We demonstrate that the internal geometric structure of a convex cone is related to its facial structure but in the infinite-dimensional setting these two structures may differ each other. Further, we study the internal geometric structure of conical halfspaces (convex cones whose complements are also convex cones). We show that every conical halfspace is the disjoint union of the linear ordered family of its open components each of which is a conical halfspace in its linear hull. Using the internal geometric structure of conical halfspaces, each asymmetric 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.