Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.