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 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/Q of the set Q by the equivalence relation Q, which is the symmetric part of the dominance relation 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 Q induces a partial order relation Q^* on the family O (Q): = Q/Q of all open components of the set Q with respect to which the partially ordered family (O (Q), Q^*) is an upper semilattice. For halfspaces (convex sets H whose complements are also convex), the corresponding upper semilattice (O (H), 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 (O (Q), 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. , O (F) O (Q). Moreover, the partial order relation F^* defined on O (F) coincides with the restriction to O (F) of the partial order relation Q^* defined on O (Q). Thus, the internal structure (O (F), F^*) of any face F of a convex set Q is a substructure of the internal structure (O (Q), Q^*) of Q itself.
V. V. Gorokhovik (Fri,) studied this question.