= Holomorphic convex hull
{title2=$\widehat K_\Omega$}
For <compact> $K$ in a plane domain $\Omega$, its <holomorphic convex hull> is
$$
\widehat K_{\Omega}=\{w\in\Omega:|f(w)|\le\sup_K|f|\text{ for every }f\in\mathcal O(\Omega)\}.
$$
It is $K$ together with the components of $\widehat{\mathbb C}\setminus K$ lying entirely in $\Omega$. Inclusion of those components follows from the <maximum modulus principle>. For exclusion, suppose $w$ lies in a complementary component meeting $\widehat{\mathbb C}\setminus\Omega$. Choose an allowed <pole> there. The open-and-closed resolvent argument in the proof of <Runge theorem> approximates $(z-w)^{-1}$ uniformly on $K$ by functions $h$ <holomorphic> on $\Omega$. Then $1-(z-w)h(z)$ has value one at $w$ and arbitrarily small modulus on $K$, separating $w$ from the hull. For $\Omega=\mathbb C$, this is the <polynomial hull in one complex variable>.
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