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  "body": "\n## Summary\n\nLet $K$ be a compact subset of an open set $D \\subseteq \\C^n$.\nIts holomorphically convex hull is the set of points of $D$ where no [[holomorphic function|507C34W0ZT]] on $D$ beats, in absolute value, its maximum on $K$.\nIt is the analogue of the convex hull, with holomorphic functions in place of linear functions.\nWhether these hulls stay away from the boundary of $D$ decides whether $D$ is a domain of holomorphy.\n\n## Explanations\n\n### L1 human\n\nThe closed [[convex hull|0FQSDC2EQ2]] of a compact set consists of the points that every closed half-space containing the set also contains.\nThe holomorphic hull replaces half-spaces by the regions $\\abs{f} \\le c$ for holomorphic functions $f$.\nFor the unit circle in $\\C$, the hull is the closed unit disc.\nBy the maximum principle, no holomorphic function on $\\C$ can be larger inside the disc than on the circle.\nIf the origin is removed from the domain, the hull of the circle is just the circle, because the function $1/z$ is larger inside.\nSo the hull depends on the domain $D$ as well as on $K$.\n",
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