Every compact convex set in $\mathbb{C}$ is the intersection of all closed half-space containing it

Question

I am working on the book "A Short course on Spectral Theory", written by William Arveson.

On page 28, there is Remark 1.10.2 stated that:

Every compact convex set $K \subseteq \mathbb{C}$ is the intersection of all closed half-space that contain it. It is also true that $K$ is the intersection of all closed disk that contain it. Equivalently, if $z_0 \in \mathbb{C}$ is any point not in the closed convex hull of $K$, then there is a disk $D = D_{a, R} = \left \{ z \in \mathbb{C} : > |z-a| \le R\right \}$ such that $K \subseteq D$ and $z_0 \neq D$

I tried to prove this remark in detail and sketch an illustration, however, I could not.

Thank you for your help.

Answer 1

Hint: If $z_0\notin K$, let $z_1\in K$ be a point in $K$ closest to $z_0$. (It is unique, but you don't need that for the proof.) Look at the normal bisector of the line segment between $z_0$ and $z_1$.