Internal structure of convex sets and their faces

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 $Q/\mathbin{<\!>}_Q$ of the set $Q$ by the equivalence relation $\mathbin{<\!>}_Q$, 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 ${\mathcal O}(Q):= Q/\mathbin{<\!>}_Q$ 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.