Piercing Hyperplane Theorem

We prove that any strictly convex and closed set in \(\mathbb{R}^n\) is an affine subspace if it contains a hyperplane as a subset. In other words, no hyperplane fits into a strictly convex and closed set \(C\) unless \(C\) is flat. We also present certain applications of this result in economic theory reminiscent of the separating and supporting hyperplane theorems.