On proper separation of convex sets

The aim of this contribution is to propose an alternative but equivalent statement of the proper separation of two closed convex sets in a finite-dimensional Euclidean space. To this aim, we characterize the affine hull of a closed convex set defined by a finite set of equalities and inequalities. Furthermore, we describe algebraically the relative interior of this set by projecting the optimal set of a convex optimization problem onto a subspace of its variables. Then we use this description to develop a system of equalities and inequalities by which the proper separability of the given convex sets is identified. We show that this system is linear in the special case that the given sets are polyhedral.