The book I am referring regarding bell inequalites has the following two things it mentions about two convex sets which I am unable to understand. There are two convex sets $C$ ($C$ is a polytope ) and $Q$. The book says
- $Q$ is a convex set but not closed. What does it mean ? Is it a convex set with infinite extreme points ?
- There is some relation because of which it comes out that $Q \subset C$. But in the next theorem the author proves $Q$ contains the interior of $C$. How is it different from the condition $Q \subset C$ ?
1) An open ball is a simple example of a convex set which is not closed. The only requirement for convexity is that a line between any two points in the set remains in the set. Clearly, that doesn't imply closure!
2) If $Q\subset C$, that doesn't imply that $Q$ contains the whole interior of $C$. If I have a ball of radius $r$, then a ball of radius $r/2$ could fit inside it. Thus the smaller ball is a subset of the larger ball. But the smaller ball certainly doesn't contain the whole interior of the larger one!