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 <math>+ \infty</math> instead of <math>0</math> to the outside elements.
''Each field seems to have its own meaning of an "indicator function", as in complex analysis for instance.''
==Definition==
Let <math>X</math> be a set, and let <math>A</math> be a subset of <math>X</math>. The '''indicator function''' of <math>A</math> is the function <ref name=rocka1>R. T. Rockafellar, '' Convex Analysis'', Princeton University Press, (1997) [1970], p.28.</ref> <ref name=jbhu1>J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization I'', Springer-Verlag, 1993, p.152.</ref> <ref name=boyd1>S. Boyd, L. Vandenberghe, ''Convex Optimization'', Cambridge University Press, (2009) [2004], p.68.</ref> <ref name=bc1>H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.12.</ref>
:<math>\iota_{A} : X \to \mathbb{R} \cup \{ + \infty \}</math>
taking values in the extended real number line defined by
:<math>\iota_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>
==Properties==
This function is convex if and only if the set <math>A</math> is convex.<ref name=bc2>H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.139.</ref>
This function is lower-semicontinuous if and only if the set <math>A</math> is closed.<ref name=bc1/>
For any arbitrary sets <math>A</math> and <math>B</math>, it is that <math>\iota_A + \iota_B = \iota_{A\cap B}</math>.
For an arbitrary non-empty set its Legendre transform is the support function.<ref name=jbhu2>J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization II'', Springer-Verlag, 1993, p.39.</ref>
The subgradient of <math>\iota_{A} (x)</math> for a set <math>A</math> and <math>x\in A</math> is the normal cone of that set at <math>x</math>.<ref name=bc3>H. H. Bauschke, P. L. Combettes, ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces'', Springer (2017) [2011], p.267.</ref>
Its infimal convolution with the Euclidean norm <math>||\cdot||_2 </math> is the Euclidean distance to that set.<ref name=jbhu3>J. B. Hiriart-Urruty, C. Lemaréchal, ''Convex Analysis and Optimization II'', Springer-Verlag, 1993, p.65.</ref>
==References== {{reflist}}
==Bibliography== * {{cite book | last = Rockafellar | first = R. T. | authorlink = R. Tyrrell Rockafellar | title = Convex Analysis | publisher = Princeton University Press | location = Princeton, NJ | year = 1997 | orig-date = 1970 | isbn = 978-0-691-01586-6 }} * {{cite book | first1 = J. B. | last1 = Hiriart-Urruty | first2 = C. | last2 = Lemaréchal | title = Convex Analysis and Minimization Algorithms I & II | publisher = Springer-Verlag | year = 1993 }} * {{cite book | first1 = S. P. | last1 = Boyd | first2 = L. | last2 = Vandenberghe | title = Convex Optimization | publisher = Cambridge University Press | year = 2004 }} * {{cite book | first1 = H. H. | last1 = Bauschke | first2 = P. L. | last2 = Combettes | title = Convex Analysis and Monotone Operator Theory in Hilbert Spaces | publisher = Springer | year = 2011 }}
Category:Convex analysis