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effective domain ：ウィキペディア英語版

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 \cup \

, has an ''effective domain'' defined by
:

\operatornamef = \. \,

If the function is concave, then the ''effective domain'' is
:

\operatornamef = \. \,

〔
The effective domain is equivalent to the projection of the epigraph of a function

f: X \to \mathbb \cup \

onto ''X''. That is
:

\operatornamef = \f\}. \,

Note that if a convex function is mapping to the normal real number line given by

f: X \to \mathbb

then the effective domain is the same as the normal definition of the domain.
A function

f: X \to \mathbb \cup \

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

.〔
== References ==

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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