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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.

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Your edit has made the question unclear to me. In the original question, you ask about a relationship between extremal points of $C$ and of $\mathcal L$. In the edit, you ask about a relationship between the image of the extremal points of $C$ under $T$ and the extremal points of the image of $C$ under $T$. Those seem to be two very different questions. Or does "what I want is" merely refer to a part of the assumptions? If so, which part? –  joriki Sep 13 '12 at 9:00
    
Thanks @joriki. It was nonsense stupid, and hence removed. What I wanted was perhaps the statement "Let $E(C)$ be the set of extremal points of $C$. Is it possible that $T$ is an extremal map if and only if $T(E(C))\subseteq E(T(C))$ (or something similar)". However, I am having a feeling that, boundary of $C$ will be mapped to the boundary of $T(C)$. Thus the conditions written will be satisfied for any $T\mathfrak{L}$. Am I right here, or again doing some mistake? –  RSG Sep 13 '12 at 15:06

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