Let $\sigma_1,\sigma_2$ be strongly convex polyhedral cones.
Is it true that $\sigma_1 \cap\sigma_2$ is always a face of both $\sigma_1$ and $\sigma_2$?
My guess is yes and here is my argument:
Let $u =0 \in (\sigma_1 \cup \sigma_2)^{\vee}$ be the 0 linear form.
$\forall \sigma_1,\sigma_2 \in \Sigma$, we have $\sigma_1\cap\sigma_2 = \sigma_1\cap \sigma_2 \cap (u|_{\sigma_1})^{\bot} = \sigma_1\cap \sigma_2 \cap (u|_{\sigma_2})^{\bot}$ so $\sigma_1 \cap \sigma_2$ is a face of $\sigma_1$ and $\sigma_2$ (at least in the context of strongly convex polyhedral cones).
Thank you for your help.