Indicator function (convex analysis)

In the field of mathematics known as convex analysis, the indicator 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 indicator function used in probability, but assigns $+\mathrm{\infty }$ instead of $1$ to the outside elements.

Each field seems to have its own meaning of an "indicator function", as in complex analysis for instance.

Definition

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

${\iota }_A:X\to ℝ\cup \{+\mathrm{\infty }\}$

taking values in the extended real number line defined by

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

Properties

This function is convex if and only if the set $A$ is convex.

This function is lower-semicontinuous if and only if the set $A$ is closed.

For any arbitrary sets $A$ and $B$, it is that ${\iota }_A+{\iota }_B={\iota }_{A\cap B}$.

For an arbitrary non-empty set its Legendre transform is the support function.

The subgradient of ${\iota }_A(x)$ for a set $A$ and $x\in A$ is the normal cone of that set at $x$.

Its infimal convolution with the Euclidean norm $||\cdot |{|}_2$ is the Euclidean distance to that set.