MathOverflow is a question and answer site for professional mathematicians. 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 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.

share|cite|improve this question
You mean that $C$ is closed but not bounded. – Denis Serre Sep 13 '12 at 16:00
@Denis Serre Yes. Example $\lbrace (x,y,z)\in\mathbb{R}^3: ~x^2+y^2\leq z^2\rbrace$ – RSG Sep 13 '12 at 16:37
up vote 2 down vote accepted

I'm not sure what counts as a "relation" here. It is tempting to think that an extreme point of $\mathfrak L$ would map at least some extreme points of $C$ to extreme points of $C$, but this is not true. For example, in ${\mathbb R}^2$ let $C$ be the convex hull of $(-1,-2)$, $(-1,2)$, $(2,-1)$ and $(2,1)$. Then one of the extreme points of $\mathfrak L$ is $\pmatrix{-1/2 & 0\cr 0 &3/4\cr}$, which maps the extreme points of $C$ to $(1/2, -3/2)$, $(1/2, 3/2)$, $(-1,-3/4)$ and $(-1,3/4)$.

If you want an example with $C$ unbounded, take the cartesian product of this with $[0,\infty)$.

share|cite|improve this answer

If $C$ is a closed, convex, pointed, solid cone, then there are some results. For example, see

  1. B. S. Tam, A geometric treatment of generalized inverses and semigroups of nonnegative matrices, Linear Algebra Appl., 41 (1981), 225-272.

  2. R. Loewy and H. Schneider, Positive operators on the $n$-dimensional ice-cream cone, J. Math. Anal. Appl., 49 (1975), 375-392.

share|cite|improve this answer

Your Answer


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

Not the answer you're looking for? Browse other questions tagged or ask your own question.