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.

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


 * $$\chi_{A} : X \to \mathbb{R} \cup \{ + \infty \}$$

taking values in the extended real number line defined by


 * $$\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}$$

Relationship with the indicator function
Let $$\mathbf{1}_{A} : X \to \mathbb{R}$$ denote the usual indicator function:


 * $$\mathbf{1}_{A} (x) := \begin{cases} 1, & x \in A; \\ 0, & x \not \in A. \end{cases}$$

If one adopts the conventions that
 * for any $$a \in \mathbb{R} \cup \{ + \infty \}$$, $$a + (+ \infty) = + \infty$$ and $$a (+\infty) = + \infty$$, except $$0(+\infty)=0$$;
 * $$\frac{1}{0} = + \infty$$; and
 * $$\frac{1}{+ \infty} = 0$$;

then the indicator and characteristic functions are related by the equations


 * $$\mathbf{1}_{A} (x) = \frac{1}{1 + \chi_{A} (x)}$$

and


 * $$\chi_{A} (x) = (+ \infty) \left( 1 - \mathbf{1}_{A} (x) \right).$$

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