$x^*\in X^*$ for which the following two inequality hold: $$\label{ps1}
\sup\big\{\la x^*,w_1\ra\;\big|\;w_1\in\Omega_1\big\}\le\inf\big\{\la x^*,w_2\ra\;\big|\;w_2\in\Omega_2\big\},$$ $$\label{ps2}
\inf\big\{\la x^*,w_1\ra\;\big|\;w_1\in\Omega_1\big\}<\sup\big\{\la x^*,w_2\ra\;\big|\;w_2\in\Omega_2\big\}.$$ Observe that condition can be equivalently rewritten as $$\la x^*,w_1\ra\le\la x^*,w_2\ra\;\mbox{\rm whenever }w_1\in\Omega_1,\;w_2\in\Omega_2,$$ while means that there exist $\bar{w}_1\in \Omega_1$ and $\bar{w}_2\in \Omega_2$ such that $$\la x^*,\bar{w}_1\ra<\la x^*,\bar{w}_2\ra.$$

Given a nonempty subset $\Omega$ of $X$, the [*relative interior*]{} of $\Omega$ is defined by $$\label{ri}
\mbox{\rm ri}(\Omega):=\big\{x\in\Omega\;\big|\;\exists\;\mbox{\rm a neighborhood}\, V\, \mbox{\rm of }x\; \mbox{\rm such that }\;V\cap\Bar{\aff}(\Omega)\subset\Omega\big\},$$ where $\Bar{\aff}(\Omega)$ denotes the closure of the affine hull of $\Omega$. If $X=\R^n$, the closure operation is not needed in since the affine hull $\aff(\Omega)$ is always closed. The following useful result, which is taken from [@bmncal Theorem 4.7], provides a proper separation description of the relative interior for convex sets in finite-dimensional spaces.

[**(relative interior and proper separation in finite dimensions).**]{}\[pst1\] Let $\Omega_1$ and $\Omega_2$ be two nonempty convex subsets of $\R^n.$ Then $\Omega_1$ and $\Omega_2$ are properly separated if and only if $\ri(\Omega_1)\cap \ri(\Omega_2)=\emptyset.$

Theorem \[pst1\] is employed below to provide further characterizations of the relative interior for convex sets. The next theorem combines various known results in this direction (see, e.g., [@r]) that are important for understanding the subsequent extensions in infinite dimensions. For 