Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 1 down vote accepted

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.