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.

## Contents

* 1 Definition
* 2 Relationship with the indicator function
* 3 Subgradient
* 4 Bibliography

## Definition

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

χA:X->ℝ∪{+∞}

taking values in the extended real number line defined by

χA(x):={0,x∈A;+∞,x∉A.

## Relationship with the indicator function

Let 𝟏A:X->ℝ denote the usual indicator function:

𝟏A(x):={1,x∈A;0,x∉A.

If one adopts the conventions that

* for any a∈ℝ∪{+∞}, a+(+∞)=+∞ and a(+∞)=+∞, except 0(+∞)=0;
* 10=+∞; and
* 1+∞=0;

then the indicator and characteristic functions are related by the equations

𝟏A(x)=11+χA(x)

and

χA(x)=(+∞)(1−𝟏A(x)).

## Subgradient

The subgradient of χA(x) for a set A is the tangent cone of that set in x.

## Bibliography

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

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