Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$

Fix $m\ge 1$ and read ${\mathbb{C}}^m$ through Complex $m$-space and its real coordinate dictionary. A polyradius is a function $r:m\to \mathbb{R}$ with $r_k>0$ for every $k<m$; a single positive real $r$ abbreviates the constant polyradius with every $r_k=r$.

For $a\in {\mathbb{C}}^m$ and a polyradius $r$, the open polydisc, the closed polydisc and the distinguished boundary are

${\mathrm{\Delta }}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<r_k\text{ for every }k<m\},$ ${\bar{\mathrm{\Delta }}}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }\le r_k\text{ for every }k<m\},$ ${\mathrm{\Gamma }}_r(a):=\{z:\mathrm{\mid }{z}_k-a_k\mathrm{\mid }=r_k\text{ for every }k<m\}\mathrm{.}$

Thus ${\mathrm{\Gamma }}_r(a)$ is the set of points all of whose coordinates lie on their own circle: it is the product of the $m$ circles $\{\mathrm{\mid }{\zeta }_k-a_k\mathrm{\mid }=r_k\}$.

The open ball and closed ball of centre $a$ and radius $\rho >0$ are those of the norm $\parallel \cdot \parallel$ of the dictionary, that is the sets $B(a,\rho )=\{z:\parallel z-a\parallel <\rho \}$ and $\bar{B}(a,\rho )=\{z:\parallel z-a\parallel \le \rho \}$ of Open ball, closed ball and sphere in a metric space and Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$.

The distinguished boundary is not the topological boundary when $m\ge 2$. The topological boundary of ${\bar{\mathrm{\Delta }}}_r(a)$ consists of the points where at least one coordinate satisfies $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }=r_k$, whereas ${\mathrm{\Gamma }}_r(a)$ requires every coordinate to do so. For $m=1$ the two coincide. For $m\ge 2$ the inclusion ${\mathrm{\Gamma }}_r(a)\subseteq \mathrm{\partial }{\bar{\mathrm{\Delta }}}_r(a)$ is proper: the point whose first coordinate is $a_0+r_0$ and whose remaining coordinates are $a_k$ lies in the topological boundary and not in ${\mathrm{\Gamma }}_r(a)$.

Polydiscs are open and convex. Openness is coordinatewise: if $\mathrm{\mid }{z}_k-a_k\mathrm{\mid }<r_k$ for every $k$, then $B(z,\rho )\subseteq {\mathrm{\Delta }}_r(a)$ for $\rho ={\min }_{k<m}(r_k-\mathrm{\mid }{z}_k-a_k\mathrm{\mid })>0$, because $\mathrm{\mid }{w}_k-z_k\mathrm{\mid }\le \parallel w-z\parallel$ by the dictionary. Convexity in the sense of A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points is also coordinatewise: for $z,w$ in ${\mathrm{\Delta }}_r(a)$ and $t\in [0,1]$, $\mathrm{\mid }(1-t)z_k+t{w}_k-a_k\mathrm{\mid }\le (1-t)\mathrm{\mid }{z}_k-a_k\mathrm{\mid }+t\mathrm{\mid }{w}_k-a_k\mathrm{\mid }<r_k$ by Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive. Hence a polydisc is star-shaped with respect to each of its points (Star-shaped open subsets of Euclidean space). The same computation gives convexity of the closed polydisc.

Slices are discs. Fixing all coordinates but the $k$th at values $a_j^{\mathrm{\prime }}$ with $\mathrm{\mid }{a}_j^{\mathrm{\prime }}-a_j\mathrm{\mid }<r_j$, the set of $\zeta$ with the resulting point in ${\mathrm{\Delta }}_r(a)$ is exactly the open disc $\{\zeta :\mathrm{\mid }\zeta -a_k\mathrm{\mid }<r_k\}$; this is what makes the one-variable theory applicable one coordinate at a time. Moduli, real and imaginary parts are those of Real and imaginary parts, complex conjugation, and modulus.