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 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 R. T. Rockafellar, Convex Analysis, Princeton University Press, (1997) , p.28. J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization I, Springer-Verlag, 1993, p.152. S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, (2009) , p.68. H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) , p.12.
taking values in the extended real number line defined by
Properties
This function is convex if and only if the set is convex.H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) , p.139.
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.J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.39.
The subgradient of for a set and is the normal cone of that set at .H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer (2017) , p.267.
Its infimal convolution with the Euclidean norm is the Euclidean distance to that set.J. B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Optimization II, Springer-Verlag, 1993, p.65.
References
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