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 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.

Definition

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

Let denote the usual indicator function:
If one adopts the conventions that
then the indicator and characteristic functions are related by the equations
and