Let $K$ be an arbitrary compact subset of domain $\Omega$.
Why is the holomorphically convex hull of $K$ is contained in the convex hull of $K$?
Holomorphically convex hull of $K$ is defined as $\hat{K}_\Omega= \{z \in \Omega: |f(z)| \leq \sup_K |f|, \forall f\in A(\Omega)\}$.
I know that convex hull of $K$ is the smallest convex set that contains $K$. How can we express this formally in this context?
Thank you!
Best Answer
We can prove by geometric reasoning in $\mathbb R^2$ that the convex hull of a compact set $K$ can be characterized as exactly the intersection of all closed balls that contain $K$.
In the complex case, consider now a point $w$ outside the convex hull of $K$. Then there must be a closed ball $\overline{B_r}(z_0)$ that contins $K$ but does not contain $w$. Now apply your definition of holomorphically convex hull to $f(z) = z-z_0$.