Let $C_1$ and $C_2$ two polyhedral cones (pointed, that is, with an only vertex on $0\in R^n$), we suppose that $\dim(C_1)=\dim(C_2)=n$. If $$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\emptyset,$$ is clear that there exist a hyperplane that separate $C_1$ with respect to $C_2$. My question is, exist a $(n-1)$-dimensional face $F$ of $C_1$ or $C_2$ such that $F$ generate a hyperplane $H$ that separate $C_1$ with respect to $C_2$?. That is, is possible that I can take as hyperplane of separation some of the hyperplanes generate by some of the face of dimension $(n-1)$ on $C_1$ or $C_2$.

My question is true when $C_1\cap C_2$ has dimension (n−1), since $C_1\cap C_2$ be including on some face of dimension (n−1) of $C_1$ or $C_2$. But when $C_1\cap C_2$ has dimension $<(n−1)$ not is clear if my question is true or not.

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 $<n−1$ then it is not clear.