# Separation of two pointed polyhedral cones using hyperplanes generated by facets

Let $$C_1$$ and $$C_2$$ two pointed (that is, with vertex in $$0$$) polyhedral cones in $$\mathbb{R}^n$$ with $$\dim(C_1)=\dim(C_2)=n$$. If $$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\emptyset,$$ then it is clear that there exists a hyperplane that separates $$C_1$$ and $$C_2$$. My question is: Does there exist a $$n-1$$-dimensional face $$F$$ of $$C_1$$ or $$C_2$$ such that $$F$$ generates a hyperplane $$H$$ that separates $$C_1$$ and $$C_2$$? That is, is it possible to take as hyperplane of separation one of the hyperplanes generated by the faces of dimension $$n-1$$ of $$C_1$$ or $$C_2$$?

The answer is yes if $$C_1\cap C_2$$ has dimension $$n−1$$, since $$C_1\cap C_2$$ is then included in some face of dimension $$n−1$$ of $$C_1$$ or $$C_2$$. But if $$C_1\cap C_2$$ has dimension $$ then it is not clear.

## 1 Answer

Yes for dimension 2 (pick the cone with larger angle; one of its sides fits). No for larger dimensions.

For a counterexample in dimension $$3$$, let $$C_1$$ be a positive orthant, and let $$C_2$$ be a negative orthant rotated by $$\pi/3$$ around $$x=y=z$$. (Note that you need to check just the faces of $$C_1$$, since the cones can be swapped by an orthogonal transform.)

There are similar examples in any higher dimension.