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 of a given element in that set. It is similar to the indicator function used in probability, but assigns instead of to the outside elements.
''Each field seems to have its own meaning of an "indicator function", as in complex analysis for instance.''

Definition

Let be a set, and let be a subset of. The indicator function of is the function
taking values in the extended real number line defined by

This function is convex if and only if the set is convex.
This function is lower-semicontinuous if and only if the set is closed.
For any arbitrary sets and, it is that.
For an arbitrary non-empty set its Legendre transform is the support function.
The subgradient of for a set and is the normal cone of that set at.
Its infimal convolution with the Euclidean norm is the Euclidean distance to that set.