Holomorphically convex hull

Summary

Let $K$ be a compact subset of an open set $D \subseteq \C^n$. Its holomorphically convex hull is the set of points of $D$ where no holomorphic function on $D$ beats, in absolute value, its maximum on $K$. It is the analogue of the convex hull, with holomorphic functions in place of linear functions. Whether these hulls stay away from the boundary of $D$ decides whether $D$ is a domain of holomorphy.