Strict Separation When One Set is Open

An open convex set can be separated from a convex set with a strict inequality gap.

Strict Separation When One Set is Open

Let $X$ be a real normed space . Let $G,\Omega\subset X$ be nonempty convex sets and assume that $G$ is open .

Corollary: There exist $x^\ast \in X^\ast$ (see dual space ) and $\beta\in\mathbb{R}$ such that

\langle x^\ast ,x\rangle < \beta \le \langle x^\ast ,y\rangle \quad \text{whenever }x\in G,\ y\in\Omega.

This follows from closed hyperplane separation under an interior condition together with the openness of $G$ , which allows a strict inequality on the $G$ side.