Closure of intersections under an interior-point condition

Theorem. Let $X$ be a normed vector space, and let $\Omega_1,\Omega_2\subset X$ be convex sets such that

Then

Context. In general, $\overline{A\cap B}\subset \overline{A}\cap\overline{B}$ can be strict. Convexity plus an interior qualification condition forces equality, which is important in convex analysis and duality.

Proof idea. Use the existence of an interior point common to both sets to “stabilize” approximations and apply interior-segment geometry to build sequences in $\Omega_1\cap\Omega_2$ approximating any point in $\overline{\Omega_1}\cap\overline{\Omega_2}$.

Remark. The conclusion remains valid under the weaker condition $\mathrm{int}(\Omega_1)\cap \Omega_2\neq\emptyset$.