I came up with the following question while learning about different norms in $\mathbb{C}^n$.
For $z=(z_1, \ldots, z_n)^T \in \mathbb{C}^n$ we consider the 1-norm: $\|z\|_1= \sum_{k=1}^n|z_k|$.
Let $B= \{z \in \mathbb{C}^n : \|z\|_1 \leq 1 \}$ be the ball of radius $1$ centered at the origin. What are the extreme points of $B$? My intuition (generalising from $\mathbb{R}^n$, where things are easier to visualize) is that they should be all $z=(z_1, \ldots, z_n)^T$ where $|z_{k'}|=1$ for exactly one $k'$ and $z_k=0$ for all other $k$. But I am not sure about this because:
(1) I thought this ball would have finitely many extreme points.
(2) I have no idea how to prove that these are indeed extreme points.
I would be very grateful if someone could explain what the situation is in $\mathbb{C}^n$ (especially (1))
Any point with $\|z\|=1$ and more than one nonzero entry is not an extreme point, because it is a convex combination of points $u(j)$ with $u(j)_j = z_j/|z_j|$, $u(j)_k = 0$ otherwise.
A point with a single nonzero entry of absolute value $1$ is indeed an extreme point. Note that all points of the unit circle in the complex plane are extreme points of the unit disk.