exhaustion function

Definition.

Let $G\subset {ℂ}^n$ be a domain and let $f:G\to ℝ$ is called an exhaustion function whenever

is relatively compact in $G$ for all $r\in ℝ$.

For example $G$ is pseudoconvex if and only if $G$ has a continuousplurisubharmonic exhaustion function.

We can also define a bounded version.

Definition.

Let $G\subset {ℂ}^n$ be a domain and let $f:G\to (-\mathrm{\infty },c]$ for some $c\in ℝ$,is called a bounded exhaustion function whenever

is relatively compact in $G$ for all .

A domain which has a bounded plurisubharmonic exhaustion function is usuallyreferred to as a hyperconvex domain. Note that not all pseudoconvexdomains have a bounded plurisubharmonic exhaustion function. For examplethe Hartogs’s triangle does not, though it does have an unbounded one.