I am attempting to find a closed convex set that has no extreme points. I realize that a closed convex set has at least one extreme point if the set does not contain a line. So my example would be a set that is exactly a line.
$S=\{(x,y)\in\mathbb{R}^2\mid x-y=0\}$
I am then having trouble justifying this, because a definition of an extreme point is that if $x=\frac{1}{2}(x_{1}+x_{2})$, with $x_{1},x_{2}\in S$, then $x=x_{1}=x_{2}$. Since for every point $y=x$, and so would every point be an extreme point.
Is this just an incorrect set? And how would you go about justifying in general that these sets do not have an extreme point?
Indeed, a line has no extreme points. If $x$ and $v$ are two distinct points of the line, then $x$ is the midpoint of $2x-v$ and $v$, so $x$ is not an extreme point.
The statement that a closed convex set that does not contain a line has at least one extreme point is true in $\mathbb R^n$, but not in an infinite-dimensional Banach space. For example, the closed unit ball of the sequence space $c_0$ has no extreme point but contains no line.