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  "body": "\n## Summary\n\nThe distinguished boundary of a [[polydisc|DJBWP7D3M8]] $P(a, r)$ in $\\C^n$ is the torus of points with $\\abs{z_j - a_j} = r_j$ for every $j$.\nIt is the set over which the [[Cauchy integral formula on a polydisc|FG5448NZ18]] integrates.\nA function holomorphic on the polydisc and continuous on its closure attains its largest modulus on this torus.\nIt is the [[Shilov boundary|JHQNT58FEH]] of the algebra of such functions, and for $n \\ge 2$ it is a small part of the topological boundary.\n\n## Explanations\n\n### L1 human\n\nThe boundary of the unit polydisc in $\\C^2$ is the set where $\\abs{z_1} = 1$ or $\\abs{z_2} = 1$.\nThe distinguished boundary is the smaller set where both $\\abs{z_1} = 1$ and $\\abs{z_2} = 1$.\nIt is a product of two circles, a torus of real dimension $2$, while the full boundary has real dimension $3$.\nFor example, $f(z_1, z_2) = z_1 z_2$ has modulus at most $1$ on the closed polydisc, and it reaches $1$ exactly on the distinguished boundary.\nSo, for holomorphic functions, this thin torus already sees the maximum.\n",
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