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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a bounded domain and let $p \\in \\partial\\Omega$.\nA *peak function* for $\\Omega$ at $p$ is a function $f \\colon \\overline{\\Omega} \\to \\C$ such that:\n\n1. $f$ is continuous on $\\overline{\\Omega}$ and [[holomorphic|507C34W0ZT]] on $\\Omega$.\n2. $f(p) = 1$.\n3. $\\abs{f(z)} \u003c 1$ for every $z \\in \\overline{\\Omega} \\setminus \\set{p}$.\n\nThe point $p$ is a *peak point* of $\\Omega$ if a peak function at $p$ exists.\n\n## Remarks\n\nExample: on the unit ball $B \\subseteq \\C^n$, let $e_1 = (1, 0, \\dots, 0)$ and $f(z) = (1 + z_1)/2$.\nThen $f(e_1) = 1$, and $\\abs{f(z)} \\le (1 + \\abs{z_1})/2 \\le 1$ on $\\overline{B}$, with equality only if $z_1 = 1$, which forces $z = e_1$.\nNon-example: on the bidisc $\\Delta^2$, the point $p = (1, 0)$ is not a peak point.\nSuppose $f$ satisfies the three clauses.\nThe function $w \\mapsto f(1, w)$ is holomorphic on the open unit disc, as a locally uniform limit of the functions $w \\mapsto f(r, w)$ for $r \\to 1$.\nIt has modulus at most $1$ and reaches the modulus $1$ at the interior point $w = 0$, so it is the constant $1$ by the maximum modulus principle, against clause 3.\nEvery boundary point of a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] with smooth boundary is a peak point.\n",
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