7
$\begingroup$

Suppose that $C$ is a bounded convex subset of $\mathbb{R}^n$ such that the optimum value, over $C$, of any linear functional is attained at some point of $C$. Does this imply that $C$ is closed? If so, is there a simple proof? If not, a counterexample would be nice.

$\endgroup$
14
$\begingroup$

No, as a counterexample take a closed "stadium" in $\mathbf{R}^2$ (the convex hull of two half-circles, I hope the word stadium makes it clear) and remove from it one of the endpoints of the half-circles. This is somehow related to the difference between extreme points and exposed points.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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