Two Notions involved:
$D$-$hull$ of $K$:
$\mathcal{H}_D(K) := \{z\in D : \vert f(z)\vert \leq \Vert f \Vert_{L^\infty(K)}$ $\forall f$ holomorphic in $D$ $\}$.
Distance between a point $x_0 \in \mathbb{C}$ and a set $A \subset \mathbb{C}$ :
$dist(z_0,A) := \inf_{z\in A} \vert z - z_0 \vert$ .
The Properties I want to prove are:
- $dist(z_0,K)=dist(z_0,\mathcal{H}_D(K))\space\space\space\forall z \in D^c$
- $\mathcal{H}_D(K)$ is the union of $K$ and the connected components of $D\setminus K$ which are relatively compact in $K$.
The second property resembles the condition of Runge Theorem. But my main issue here is not able to find a way to have a geometrical interpretation of $\mathcal{H}_D(K)$.
I found an approach in Complex Analysis: A Functional Analytic Approach by Haslinger p126 : https://books.google.ca/books?id=bklADwAAQBAJ&pg=PA125&lpg=PA125&dq=Holomorphically+convex+hull+connected+components&source=bl&ots=z9B8xDq-BY&sig=ACfU3U1WKbFMEmNdBBP0BWNNKPogoZZeqw&hl=fr&sa=X&ved=2ahUKEwjmvun96tjgAhVus1kKHTfTA_EQ6AEwB3oECAMQAQ#v=onepage&q=Holomorphically%20convex%20hull%20connected%20components&f=false