Separating a Point from a Convex Set via the Core

Let $X$ be a real vector space, let $x_0\in X$, and let $\Omega\subset X$ be convex. Assume that core(Ω) is nonempty and $x_0\notin \operatorname{core}(\Omega)$.

Theorem: The sets $\Omega$ and $\{x_0\}$ can be separated by a hyperplane. Equivalently, there exists a nonzero linear functional $f:X\to\mathbb{R}$ such that

Context: This is a geometric form of Hahn–Banach. The proof uses the Minkowski gauge of $\Omega$ (after translation) to build a sublinear domination bound and then applies Hahn–Banach.