exhaustion function


Definition.

Let Gn be a domain and let f:G is called an exhaustion function whenever

{zGf(z)<r}

is relatively compact in G for all r.

For example G is pseudoconvex if and only if G has a continuousMathworldPlanetmath plurisubharmonic exhaustion function.

We can also define a bounded version.

Definition.

Let Gn be a domain and let f:G(-,c] for some c, is called a bounded exhaustion function whenever

{zGf(z)<r}

is relatively compact in G for all r<c.

A domain which has a bounded plurisubharmonic exhaustion function is usually referred to as a hyperconvex domain. Note that not all pseudoconvex domains have a bounded plurisubharmonic exhaustion function. For example the Hartogs’s triangle does not, though it does have an unbounded one.

References

  • 1 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title exhaustion function
Canonical name ExhaustionFunction
Date of creation 2013-03-22 14:32:41
Last modified on 2013-03-22 14:32:41
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 5
Author jirka (4157)
Entry type Definition
Classification msc 32U10
Classification msc 32T35
Related topic Pseudoconvex
Defines bounded exhaustion function
Defines hyperconvex