Characteristic function (convex analysis)

In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.

DefinitionEdit

Let   be a set, and let   be a subset of  . The characteristic function of   is the function

 

taking values in the extended real number line defined by

 

Relationship with the indicator functionEdit

Let   denote the usual indicator function:

 

If one adopts the conventions that

  • for any  ,   and  , except  ;
  •  ; and
  •  ;

then the indicator and characteristic functions are related by the equations

 

and

 

BibliographyEdit

  • Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.