Characteristic function (convex analysis)

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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.


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 function[edit]

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