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  "body": "\n## Statement\n\nLet $D \\subseteq \\C^n$ be a nonempty connected open set, and let $f$ be a [[holomorphic function|507C34W0ZT]] on $D$.\nIf there is a point $a \\in D$ with $\\abs{f(z)} \\le \\abs{f(a)}$ for all $z \\in D$, then $f$ is constant on $D$.\n\n## Remarks\n\nFor $z$ near $a$, the map $t \\mapsto f(a + t(z - a))$ is a holomorphic function of one variable on a disc around $t = 0$ whose modulus has a maximum at $t = 0$.\nBy the one-variable [[maximum modulus principle|048F2P90WR]] it is constant, so $f$ is constant near $a$, and the [[identity theorem|7GZ908QZ04]] makes $f$ constant on $D$.\nConsequence: if $D$ is bounded and $f$ is continuous on the closure $\\overline{D}$ and holomorphic on $D$, then $\\max_{\\overline{D}} \\abs{f}$ is attained on the boundary of $D$.\nConnectedness is needed: on two disjoint balls, a function equal to $0$ on one and $1$ on the other attains its maximum modulus without being constant.\nExample: on the unit polydisc of $\\C^2$, $f(z) = z_1 z_2$ satisfies $\\abs{f} \u003c 1$ inside and reaches $\\abs{f} = 1$ only where $\\abs{z_1} = \\abs{z_2} = 1$, on the distinguished boundary.\n\n## Explanations\n\n### L1 human\n\nThe size $\\abs{f}$ of a non-constant holomorphic function never has a highest point inside its domain.\nSo on a bounded domain the largest value of $\\abs{f}$ is found on the edge.\nIt is the same principle as in one variable, and it is proved by restricting to complex lines.\n\n### L2 human\n\nThe proof is a typical way to pass from one to several variables: restrict to complex lines through a point and use the one-variable result.\nFor special domains the maximum is found on a much smaller part of the boundary.\nOn a polydisc it is the [[distinguished boundary|A6HN207Y2T]], an $n$-dimensional torus.\nThe study of these smallest sets leads to the Shilov boundary and to peak points.\n",
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