Abstract
Most often, the geometric structure of convex sets is associated with their facial structure. In the first section of this paper, we present a somewhat different approach to characterizing the geometric structure of convex sets based on the concept of an open component of a convex set. In this paper, we consider convex sets in infinite-dimensional real vector spaces endowed with no topology. To define the notion of an open component of a convex set \(Q\) , the preorder relation \(\unlhd_Q\) is introduced on \(Q\) (its own for each set \(Q\) ) called a dominance relation. Open components of a convex set \(Q\) are defined as equivalence classes of the quotient set \(Q/{<\negthinspace >}_Q\) of the set \(Q\) by the equivalence relation \({<\negthinspace >}_Q\) , which is the symmetric part of the dominance relation \(\unlhd_Q\) . Each open component of a convex set \(Q\) is a relatively algebraic open subset of the set \(Q\) under consideration, and the set \(Q\) is a disjoint union of all open components belonging to \(Q\) . The dominance relation \(\unlhd_Q\) induces a partial order relation \(\unlhd_Q^*\) on the family \({\mathcal O}(Q):= Q/{<\negthinspace >}_Q\) of all open components of the set \(Q\) with respect to which the partially ordered family \(({\mathcal O}(Q),\unlhd_Q^*)\) is an upper semilattice. For halfspaces (convex sets \(H\) whose complements are also convex), the corresponding upper semilattice \(({\mathcal O}(H),\unlhd_H^*)\) is a linearly ordered set. The internal structure of a convex set \(Q\) is identified in the paper with the structure of the upper semilattice \(({\mathcal O}(Q),\unlhd_Q^*)\) . In the second section of the paper, the connection between the internal structure of a convex set and that of its faces is investigated. It is established that each open component of a convex set \(Q\) is a relative algebraic interior of the minimal (with respect to inclusion) face of \(Q\) containing the given open component. Conversely, if a face \(F\) of a convex set \(Q\) has a nonempty relative algebraic interior, then it (the relative algebraic interior of the face) coincides with some open component of the set \(Q\) , and the face \(F\) itself is a minimal face containing this open component (such faces are called minimal in the paper). In finite-dimensional vector spaces, any face \(F\) of a convex set \(Q\) is minimal, whereas in any infinite-dimensional vector space, there exist convex sets whose faces are not all minimal. Concurrently, each open component of any face \(F\) of a convex set \(Q\) is an open component of \(Q\) itself; i.e., \({\mathcal O}(F) \subset {\mathcal O}(Q)\) . Moreover, the partial order relation \(\unlhd_F^*\) defined on \({\mathcal O}(F)\) coincides with the restriction to \({\mathcal O}(F)\) of the partial order relation \(\unlhd_Q^*\) defined on \({\mathcal O}(Q)\) . Thus, the internal structure \(({\mathcal O}(F),\unlhd_F^*)\) of any face \(F\) of a convex set \(Q\) is a substructure of the internal structure \(({\mathcal O}(Q),\unlhd_Q^*)\) of \(Q\) itself.
