On Proper Preordering of Convex Sets

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