# If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?

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.

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.
• Just to make sure I am understanding you correctly, you take $q$ to be in the interior of $C_2$ and not in $C_1$ and then consider the line segment between $p$ and $q$. Oct 29 '20 at 19:29
• not necessarily interior, $q$ may be a boundary point of $C_2$, the whole interval $pq$ (without endpoints) still consists of interior points of $C_2$ Oct 29 '20 at 20:14