This is from Fulton's toric vareity section 1.2.
In particular, I want to show if $\tau$ is a convex subset of a (strongly convex polyhedra) cone $\sigma$ such that $u,v\in \sigma, u+v\in\tau$ implies $u,v\in\tau$ then $\tau$ is in fact a face of $\sigma$.
Here we say $\tau$ is a face of $\sigma$ iff we can find $u\in \sigma^\vee:=\{u\in \mathbb R^n:\forall v\in\sigma, \langle u,v \rangle\geq 0\}$ so that $\tau=\sigma\cap H_u$ with $H_u:=\{w\in \mathbb R^n: \langle u,w \rangle=0\}$. I failed to see how this is true by consider the following example:
Can someone please tell me what I did wrong and how to actually prove this claim? Thanks in advance.