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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be a domain, that is, a nonempty connected open set.\nThe following are equivalent.\n\n1. $D$ is a [[domain of holomorphy|4487CSC0MB]].\n2. $D$ is [[holomorphically convex|D5J6CP5V2Z]]: for every compact set $K \\subseteq D$, its [[holomorphically convex hull|CD6SMMZSW8]] $\\hat K_D$ is compact.\n3. There is a [[holomorphic function|507C34W0ZT]] $f$ on $D$ that cannot be continued beyond $D$. That is, there are no nonempty open sets $U_1 \\subseteq D \\cap U_2$, with $U_2$ connected and $U_2 \\not\\subseteq D$, and no holomorphic $g$ on $U_2$ with $g = f$ on $U_1$.\n\n## Remarks\n\nClause 3 implies clause 1 directly from the definition.\nFor 1 implies 2, the key estimate concerns a domain of holomorphy $D$.\nThe distance from $\\hat K_D$ to the boundary of $D$ equals the distance from $K$ to the boundary, measured with polydiscs.\nIt comes from the Cauchy estimates: the Taylor series at points of $\\hat K_D$ of every holomorphic function on $D$ converge on polydiscs of a fixed radius.\nFor 2 implies 3, take a sequence of points of $D$ that accumulates at every boundary point.\nHolomorphic convexity gives holomorphic functions that are large at chosen points of the sequence.\nAn infinite product or series of them gives a function unbounded near every boundary point.\n\nExample: a convex domain is holomorphically convex, because $\\abs{e^{\\ell}} = e^{\\operatorname{Re} \\ell}$ for complex linear $\\ell$ shows that $\\hat K_D$ lies in the convex hull of $K$; so it is a domain of holomorphy.\n\n## Explanations\n\n### L1 human\n\nThere are three ways to say that a domain has no hidden extensions.\nOne can ask that not all functions extend across any boundary point, or that one single function refuses to extend anywhere, or a convexity condition defined with holomorphic functions.\nThe theorem says these three conditions are the same.\n\n### L2 human\n\nHolomorphic convexity is an intrinsic condition, stated without reference to points outside $D$.\nThis makes it the right condition to generalise to complex manifolds, where it leads to the notion of a Stein manifold.\nIt reduces the Levi problem to showing that pseudoconvex domains are holomorphically convex.\nFor $n = 1$ every domain satisfies all three conditions.\n",
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