# maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ linear maps such that }L(S)\subseteq S \}$. Clearly, this is a convex set.

My question(s): Does there exist any relation between extremal points of $S$ and extremal points of $\mathcal{E}(S)$? If any such relation exists, how far we can stretch it, for example does this work for any Banach space or what if we replace linear by affine maps etc.

Does it help if we also assume $S$ to be bounded as well?

I do not know much about convex geometry. Advanced thanks for any help suggestion, reference etc. Feel free to comment and edit for any improvement and or clarity.

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