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 to the outside elements.

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

Definition

be a set be a subset. The indicator function is the function

taking values in the extended real number line defined by

Properties

is convex.

is closed.

.

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

is the normal cone.

Its infimal convolution is the Euclidean distance to that set.

References

Bibliography