The equal-radius polydisc boundary function

Let $\mathrm{\Omega }\subseteq {\mathbb{C}}^m$ be a domain and let $a\in \mathrm{\Omega }$. The equal-radius polydisc boundary function of $\mathrm{\Omega }$ at $a$ is

${\delta }_{\mathrm{\Omega }}(a):=\sup \{r>0:{\mathrm{\Delta }}_r(a)\subseteq \mathrm{\Omega }\}\in (0,+\mathrm{\infty }],$

where ${\mathrm{\Delta }}_r(a)$ is the open polydisc of constant polyradius $r$ from Balls, polydiscs and the distinguished boundary in ${\mathbb{C}}^m$.

For $E\subseteq \mathrm{\Omega }$, define

${\delta }_{\mathrm{\Omega }}(E):={\inf }_{a\in E}{\delta }_{\mathrm{\Omega }}(a)\mathrm{.}$

The infimum is taken in the extended nonnegative reals, with ${\delta }_{\mathrm{\Omega }}(\mathrm{\varnothing }):=+\mathrm{\infty }$.

Because $\mathrm{\Omega }$ is open, every point $a\in \mathrm{\Omega }$ has some positive-radius polydisc inside $\mathrm{\Omega }$, so ${\delta }_{\mathrm{\Omega }}(a)>0$.

The quantity ${\delta }_{\mathrm{\Omega }}(a)$ is the distance from $a$ to the complement of $\mathrm{\Omega }$ measured in the sup norm on coordinates, written in the language of equal-radius polydiscs because that is the form used by the several-variable Cauchy estimates.