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 1 {\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 X {\displaystyle X} be a set, and let A {\displaystyle A} be a subset of X {\displaystyle X}. The indicator function of A {\displaystyle A} is the function

ι A : X → R ∪ { + ∞ } {\displaystyle \iota _{A}:X\to \mathbb {R} \cup \{+\infty \}}

taking values in the extended real number line defined by

ι A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A . {\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 A {\displaystyle A} is convex.

This function is lower-semicontinuous if and only if the set A {\displaystyle A} is closed.

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

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

The subgradient of ι A ( x ) {\displaystyle \iota _{A}(x)} for a set A {\displaystyle A} and x ∈ A {\displaystyle x\in A} is the normal cone of that set at x {\displaystyle x}.

Its infimal convolution with the Euclidean norm | | ⋅ | | 2 {\displaystyle ||\cdot ||_{2}} is the Euclidean distance to that set.

Bibliography

  • Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.
  • Hiriart-Urruty, J. B.; Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms I & II. Springer-Verlag.
  • Boyd, S. P.; Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
  • Bauschke, H. H.; Combettes, P. L. (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer.