Distinguished boundary of a polydisc

Remarks

The set $T(a, r)$ is the product of the $n$ circles $\abs{z_j - a_j} = r_j$, so it is a real torus of dimension $n$. It is also called the skeleton of the polydisc, and it is written $\partial_0 P(a, r)$ in some texts. Example: $(1, -i)$ lies in the distinguished boundary of the unit polydisc of $\C^2$, since $\abs{1} = 1$ and $\abs{-i} = 1$. Non-example: $(1, 0)$ lies in the topological boundary of the unit polydisc but not in its distinguished boundary, since $\abs{0} \ne 1$. For $n = 1$ the distinguished boundary is the whole boundary circle. If $f$ 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 $T(a, r)$.