Separating a Point from a Convex Set via the Core

If x0 is outside core(Ω) and core(Ω)≠∅, then Ω and {x0} are separable by a hyperplane.

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

f(x)\le f(x_0)\quad\text{for all }x\in\Omega.

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.