Let $C$ be a convex cone and $C^*$ its dual cone. It seems for me that either $C\subseteq C^*$ or $C^* \subseteq C$ at least in 2 dimension. Is it correct? if so, is it the case also for higher dimension?
I'm interested in the proof if it is true or a contradictory example if it is not.
Thanks.
On
One good example of a convex cone to keep in mind is a subspace $C$ of $\mathbb R^n$. In this case, the dual cone $C^*$ is the orthogonal complement of $C$. So there is no inclusion either way in this example.
Here's a proof that if $C$ is a subspace of $\mathbb R^n$ then $C^* = C^\perp$ (the orthogonal complement of $C$). First of all, clearly $C^\perp \subset C^*$. So we only need to show that $C^* \subset C^\perp$. Let $x \in C^*$. If $y \in C$, then $\langle x , y \rangle \geq 0$ and also $\langle x , -y \rangle \geq 0$. Thus $\langle x, y \rangle = 0$. This shows that $x$ is orthogonal to every vector in $C$. In other words, $x \in C^\perp$. So, we can conclude that $C^* \subset C^\perp$.
I think this is correct in two dimensions (if $\operatorname{span}(C)$ is dense) and a proof should be not too hard. However, there are counterexamples in dimension $3$, see https://math.stackexchange.com/a/1482741/58577.