Reinhardt domain

Remarks

Example: the unit polydisc $\set{\abs{z_1} < 1, \abs{z_2} < 1}$ is a complete Reinhardt domain, since both conditions only involve $\abs{z_1}$ and $\abs{z_2}$ and stay true when these decrease. The annulus product $\set{1 < \abs{z_1} < 2, \ 1 < \abs{z_2} < 2}$ is a Reinhardt domain that is not complete: it contains $(3/2, 3/2)$ but not $(0, 3/2)$. Non-example: the ball $\set{z \in \C^2 : \abs{z_1 - 1/2}^2 + \abs{z_2}^2 < 1}$ contains $(7/5, 0)$, but its rotation $(-7/5, 0)$ is at distance $19/10 > 1$ from the centre, so it is not Reinhardt. A Reinhardt domain with centre $a$ is a set $a + D$ with $D$ a Reinhardt domain with centre $0$. A Reinhardt domain is determined by its image under $z \mapsto (\abs{z_1}, \dots, \abs{z_n})$.