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.



Characterization of closed convex set
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.


