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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.

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What are the arguments that make you say "it's not hard to see why it must be true"? –  Davide Giraudo Jan 12 '12 at 20:53
    
@Davide My statement, that you quote, is not precise and is not related to the problem. It's just intuition based on a number of particular cases. –  student Jan 12 '12 at 22:16

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I think that this question is answered by Theorem 4 in A. Ben-Israel, Notes on linear inequalities, 1: The intersection of the nonnegative orthant with complementary orthogonal subspaces, J. Math. Anal. Appl. 9:303-314 (1964).

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