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  "body": "\n## Definition\n\nA *domain* in $\\C^n$ is a nonempty [[open|02BH3MCBQB]] and [[connected|0V032VWTHD]] subset of $\\C^n$.\nA domain $D \\subseteq \\C^n$ is a *Reinhardt domain* (with centre $0$) if for every $z \\in D$ and every $\\theta = (\\theta_1, \\dots, \\theta_n) \\in \\R^n$ the point\n$$(e^{i\\theta_1} z_1, \\dots, e^{i\\theta_n} z_n)$$\nlies in $D$.\nA Reinhardt domain $D$ is *complete* if for every $z \\in D$ and every $w \\in \\C^n$ with $\\abs{w_j} \\le \\abs{z_j}$ for $j = 1, \\dots, n$, the point $w$ lies in $D$.\n\n## Remarks\n\nExample: the unit polydisc $\\set{\\abs{z_1} \u003c 1, \\abs{z_2} \u003c 1}$ is a complete Reinhardt domain, since both conditions only involve $\\abs{z_1}$ and $\\abs{z_2}$ and stay true when these decrease.\nThe annulus product $\\set{1 \u003c \\abs{z_1} \u003c 2, \\ 1 \u003c \\abs{z_2} \u003c 2}$ is a Reinhardt domain that is not complete: it contains $(3/2, 3/2)$ but not $(0, 3/2)$.\nNon-example: the ball $\\set{z \\in \\C^2 : \\abs{z_1 - 1/2}^2 + \\abs{z_2}^2 \u003c 1}$ contains $(7/5, 0)$, but its rotation $(-7/5, 0)$ is at distance $19/10 \u003e 1$ from the centre, so it is not Reinhardt.\nA Reinhardt domain with centre $a$ is a set $a + D$ with $D$ a Reinhardt domain with centre $0$.\nA Reinhardt domain is determined by its image under $z \\mapsto (\\abs{z_1}, \\dots, \\abs{z_n})$.\n",
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