In $\mathbb{R}^n$, consider the closed cone $$C^+ = \{ (x_1, \ldots, x_n) : x_i \geq 0,~~i= 1, \ldots, n\}.$$ Let $S \subseteq \mathbb{R}^n$ be a subspace (of any dimension) such that $S \cap C^+ = \{0\}$. Prove that $S^{\perp}$ has non-empty intersection with the interior of $C^+$.
The orthogonal complement is taken with respect to the canonical inner product.
It's not hard to see why this must be true, but a real proof has eluded me for some time now.