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.

Let ${\displaystyle X}$ be a set, and let ${\displaystyle A}$ be a subset of ${\displaystyle X}$. The characteristic function of ${\displaystyle A}$ is the function

${\displaystyle {\chi }_A:X\to \mathbb{ℝ}\cup \{+\mathrm{\infty }\}}$

taking values in the extended real number line defined by

${\displaystyle {\chi }_A(x):=\{\begin{array}{ll}0,& x\in A;\\ +\mathrm{\infty },& x\in ̸A.\end{array}}$

Let ${\displaystyle {\mathbf{𝟏}}_A:X\to \mathbb{ℝ}}$ denote the usual indicator function:

${\displaystyle {\mathbf{𝟏}}_A(x):=\{\begin{array}{ll}1,& x\in A;\\ 0,& x\in ̸A.\end{array}}$

If one adopts the conventions that

• for any ${\displaystyle a\in \mathbb{ℝ}\cup \{+\mathrm{\infty }\}}$, ${\displaystyle a+(+\mathrm{\infty })=+\mathrm{\infty }}$ and ${\displaystyle a(+\mathrm{\infty })=+\mathrm{\infty }}$, except ${\displaystyle 0(+\mathrm{\infty })=0}$;
• ${\displaystyle \frac{1}{0}=+\mathrm{\infty }}$; and
• ${\displaystyle \frac{1}{+\mathrm{\infty }}=0}$;

then the indicator and characteristic functions are related by the equations

${\displaystyle {\mathbf{𝟏}}_A(x)=\frac{1}{1+{\chi }_A(x)}}$

and

${\displaystyle {\chi }_A(x)=(+\mathrm{\infty })(1-{\mathbf{𝟏}}_A(x)).}$

The subgradient of ${\displaystyle {\chi }_A(x)}$ for a set ${\displaystyle A}$ is the tangent cone of that set in ${\displaystyle x}$.

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

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