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 ${\displaystyle +\infty }$ instead of ${\displaystyle 1}$ to the outside elements.

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

Definition

Let ${\displaystyle X}$ be a set, and let ${\displaystyle A}$ be a subset of ${\displaystyle X}$. The indicator function of ${\displaystyle A}$ is the function ^{[1] [2] [3] [4]}

${\displaystyle \iota _{A}:X\to \mathbb {R} \cup \{+\infty \}}$

taking values in the extended real number line defined by

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

Properties

This function is convex if and only if the set ${\displaystyle A}$ is convex.^[5]

This function is lower-semicontinuous if and only if the set ${\displaystyle A}$ is closed.^[4]

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

For an arbitrary non-empty set its Legendre transform is the support function.^[6]

The subgradient of ${\displaystyle \iota _{A}(x)}$ for a set ${\displaystyle A}$ and ${\displaystyle x\in A}$ is the normal cone of that set at ${\displaystyle x}$.^[7]

Its infimal convolution with the Euclidean norm ${\displaystyle ||\cdot ||_{2}}$ is the Euclidean distance to that set.^[8]