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# Effective domain

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 Title: Effective domain Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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, f: X \to \mathbb{R} \cup \{\pm \infty\}, has an effective domain defined by

\operatorname{dom}f = \{x \in X: f(x) < +\infty\}. \,

If the function is concave, then the effective domain is

\operatorname{dom}f = \{x \in X: f(x) > -\infty\}. \,

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

\operatorname{dom}f = \{x \in X: \exists y \in \mathbb{R}: (x,y) \in \operatorname{epi}f\}. \,

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 \in X.