In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function.
Given a vector space X then a convex function mapping to the extended reals, \( f:X\to {\mathbb {R}}\cup \{\pm \infty \} \), has an effective domain defined by
\( {\displaystyle \operatorname {dom} f=\{x\in X:f(x)<+\infty \}.} \) [1][2]
If the function is concave, then the effective domain is
\( {\displaystyle \operatorname {dom} f=\{x\in X:f(x)>-\infty \}.} \) [1]
The effective domain is equivalent to the projection of the epigraph of a function \( f:X\to {\mathbb {R}}\cup \{\pm \infty \} \) onto X. That is
\( {\displaystyle \operatorname {dom} f=\{x\in X:\exists y\in \mathbb {R} :(x,y)\in \operatorname {epi} f\}.} \) [3]
Note that if a convex function is mapping to the normal real number line given by \( f:X\to {\mathbb {R}} \) then the effective domain is the same as the normal definition of the domain.
A function \( f:X\to {\mathbb {R}}\cup \{\pm \infty \} \) is a proper convex function if and only if f is convex, the effective domain of f is nonempty and \( f(x)>-\infty \) for every x ∈ X {\displaystyle x\in X} x\in X.[3]
References
Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 254. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. p. 400. ISBN 978-3-11-018346-7.
Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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