Let $C\_1$ and $C\_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C\_1\subseteq C\_2$ and that the boundary of $C\_1$ is contained in the boundary of $C\_2$. Then is $C\_1=C\_2$? Any references for a result of this form would be welcome. I suspect this to be true, and I have some rough ideas about how to prove this, but my arguments seem messy and I am worried that my intuition from low dimensional and finitely generated cases goes wrong in the generality I am considering.


By pointed I mean that $(-C)\cap C=\{0\}$ and by full dimensional I mean that $C$ spans $\mathbb{R}^n$. I am particularly interested when $C\_1,C\_2$ are not necessarily finitely generated, though partial results in the finitely generated or finitely generated rational case would be welcome.



If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?
Assume that $q\in C\_2\setminus C\_1$. Let $p$ be an interior point of $C\_1$. Then the interval $(p,q)$ contains a boundary point of $C\_1$ but only interior points of $C\_2$. A contradiction.


