Suppose I have a convex set $C\subset\mathbb{R}^n$ such that $0\in C$ and 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}=\lbrace T:\mathbb{R}^n\rightarrow \mathbb{R}^n, \text{ $T$ is linear}, T(C)\subseteq C\rbrace\end{equation} Is there any relation between extremal points (as well as exposed points) of $C$ and extremal points (exposed points) of $\mathfrak{L}$?

I do not work in convex geometry, and so do not know whether the statement is making sense. I have asked this question here, but did not get any response (and so I decided to ask it here). Please give suggestions and feel free to correct (and edit as well), if I am wrong and/or there is some ambiguity. Advanced thanks for any help.