Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ ad every Cauchy sequence in $C$ converges in $C$, but $C$ need not be bounded. (Actually I want unbounded $C$). Consider the set \begin{equation}\mathfrak{L}=\{T:\mathbb{R}^n\rightarrow \mathbb{R}^n, \text{ $T$ is linear}, T(C)\subseteq C\}\end{equation} Is there any relation between extremal points (and rays, faces, well any such thing) of $C$ and extremal points of $\mathfrak{L}$?
I do not work in convex geometry, and so do not know whether the statement is making sense. Please give suggestions and feel free to correct (and edit), if I am wrong. Advanced thanks for any help.