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Effective domain

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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\}. \,[1][2]

If the function is concave, then the effective domain is

\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

\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 \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.  
  2. ^ Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. p. 400.  
  3. ^ a b  


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