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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]] and let $K \\subseteq D$ be a [[compact set|0R6EWFKRCF]].\nThe *holomorphically convex hull* of $K$ in $D$ is\n$$\\hat K_D = \\set{z \\in D : \\abs{f(z)} \\le \\sup_{w \\in K} \\abs{f(w)} \\text{ for every holomorphic function } f \\text{ on } D},$$\nwhere holomorphic means [[holomorphic|507C34W0ZT]] in the sense of several variables.\n\n## Remarks\n\nAlways $K \\subseteq \\hat K_D$, and $\\hat K_D$ is closed in $D$.\nUsing $f(z) = z_j$ shows that $\\hat K_D$ is bounded, and using $f = e^{\\ell}$ for complex linear $\\ell$, with $\\abs{e^{\\ell}} = e^{\\operatorname{Re} \\ell}$, shows that $\\hat K_D$ lies in the convex hull of $K$.\n\nExample: $D = \\C$ and $K = \\set{\\abs{z} = 1}$.\nBy the maximum principle every entire $f$ satisfies $\\abs{f(z)} \\le \\max_K \\abs{f}$ for $\\abs{z} \\le 1$, and $f(z) = z$ excludes the points with $\\abs{z} \u003e 1$; so $\\hat K_D$ is the closed unit disc.\n\nWith the same $K$ and $D = \\C \\setminus \\set{0}$, the function $f(z) = 1/z$ is holomorphic on $D$ and has $\\abs{f(z)} \u003e 1 = \\max_K \\abs{f}$ for $0 \u003c \\abs{z} \u003c 1$; so here $\\hat K_D = K$.\n\nWhen $D = \\C^n$, the hull coincides with the polynomially convex hull, since entire functions are uniform limits of polynomials on compact sets.\n",
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