On Proper Preordering of Convex Sets

The paper deals with convex sets of infinite-dimensional real vector spaces

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.