Does the closed unit ball of $C(E)$ have no extreme points?

Question

Let $E$ be a bounded closed set in $\mathbb R^n$. Does the closed unit ball of $C(E)$ (the space of continuous functions on $E$ with supremum norm) have no extreme points?

Answer 1

For every nonempty compact space $X$, the closed unit ball of the Banach space $C(X)$ has extreme points. Namely, every continuous function $f: X\to \mathbb K$ such that $|f|\equiv 1$ is an extreme point (here $\mathbb K$ stands for real or complex field).

Assuming the space is normal, one can show that the above are all extreme points of the closed unit ball. Indeed, if $|f(x_0)|<1$, then $|f|<1$ is a neighborhood of $x_0$. By Urysohn's lemma, there is a nonzero continuous function $g$ supported on the set $|f|\le 1-\epsilon$. Then $\|f+t g\|\le 1$ for all $t$ sufficiently close to $0$.

As a special case, the preceding paragraph applies to a closed bounded set in $\mathbb R$, because it's a metric space.