On Proper Preordering of Convex Sets

The paper deals with convex sets of infinite-dimensional real vector spaces X X with no topology. We endow each convex set Q \subset X Q ⊂ X with a preorder relation \unlhd_Q ⊴ Q , called the dominance relation, the definition of which is based on the geometric properties of each particular set Q Q , and therefore, the dominance relation \unlhd_Q ⊴ Q is specific to each Q Q . The results presented in the paper demonstrate that the view of a convex set Q Q as a preordered set (Q,\unlhd_Q) ( Q , ⊴ Q ) can be effectively used in studying the geometric structure of both the convex set Q Q itself and its faces.