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  "body": "\n## Definition\n\nLet $P(a, r)$ be the [[polydisc|DJBWP7D3M8]] in $\\C^n$ with centre $a = (a_1, \\dots, a_n)$ and polyradius $r = (r_1, \\dots, r_n)$, each $r_j \u003e 0$.\nIts *distinguished boundary* is the set\n$$T(a, r) = \\set{z \\in \\C^n : \\abs{z_j - a_j} = r_j \\text{ for } j = 1, \\dots, n}.$$\n\n## Remarks\n\nThe set $T(a, r)$ is the product of the $n$ [[circles|060W5CBZP6]] $\\abs{z_j - a_j} = r_j$, so it is a real torus of dimension $n$.\nIt is also called the *skeleton* of the polydisc, and it is written $\\partial_0 P(a, r)$ in some texts.\nExample: $(1, -i)$ lies in the distinguished boundary of the unit polydisc of $\\C^2$, since $\\abs{1} = 1$ and $\\abs{-i} = 1$.\nNon-example: $(1, 0)$ lies in the topological boundary of the unit polydisc but not in its distinguished boundary, since $\\abs{0} \\ne 1$.\nFor $n = 1$ the distinguished boundary is the whole boundary circle.\nIf $f$ is continuous on the closed polydisc and holomorphic on $P(a, r)$, then $\\max \\abs{f}$ over the closed polydisc equals $\\max \\abs{f}$ over $T(a, r)$.\n",
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