Separation of Two Convex Sets via the Core Condition

If core(Ω1)≠∅ and core(Ω1) is disjoint from Ω2, then Ω1 and Ω2 are separable by a hyperplane.

Separation of Two Convex Sets via the Core Condition

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

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

Theorem: The sets $\Omega_1$ and $\Omega_2$ can be separated by a hyperplane .

Context: This strengthens the disjointness condition by requiring only that $\Omega_2$ avoid the core of $\Omega_1$ . The argument uses idempotence of core and the auxiliary separation lemma .