# Effective domain

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, ${\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}$, 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 ${\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}$ 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 ${\displaystyle f:X\to \mathbb {R} }$ then the effective domain is the same as the normal definition of the domain.

A function ${\displaystyle f:X\to \mathbb {R} \cup \{\pm \infty \}}$ is a proper convex function if and only if f is convex, the effective domain of f is nonempty and ${\displaystyle f(x)>-\infty }$ for every ${\displaystyle x\in X}$.[3]

## References

1. ^ a b 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.
2. ^ 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.
3. ^ a b Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. p. 23. ISBN 978-0-691-01586-6.