Separation by Closed Hyperplane Under an Interior Condition

If int(Ω1)≠∅ and int(Ω1)∩Ω2=∅, then Ω1 and Ω2 are separable by a continuous functional.

Separation by Closed Hyperplane Under an Interior Condition

Let $X$ be a real normed space and let $\Omega_1,\Omega_2\subset X$ be nonempty convex sets . Assume that int(Ω₁) $\neq\emptyset$ and

\operatorname{int}(\Omega_1)\cap\Omega_2=\emptyset.

Theorem: The sets $\Omega_1$ and $\Omega_2$ can be separated by a closed hyperplane ; i.e., there exists $x^\ast \in X^\ast \setminus\{0\}$ such that

\langle x^\ast ,x\rangle \le \langle x^\ast ,y\rangle\quad \text{for all }x\in\Omega_1,\ y\in\Omega_2.

Context: In vector spaces, the analogous separation uses possibly discontinuous linear functionals (see core-based separation ). The interior assumption ensures the separating functional is continuous, using closed level-set continuity .