Effective domain

In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function. 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 → R ∪ { ± ∞ } {displaystyle f:X o mathbb {R} cup {pm infty }} , has an effective domain defined by If the function is concave, then the effective domain is The effective domain is equivalent to the projection of the epigraph of a function f : X → R ∪ { ± ∞ } {displaystyle f:X o mathbb {R} cup {pm infty }} onto X. That is Note that if a convex function is mapping to the normal real number line given by f : X → R {displaystyle f:X o mathbb {R} } then the effective domain is the same as the normal definition of the domain. A function f : X → R ∪ { ± ∞ } {displaystyle f:X o 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 ) > − ∞ {displaystyle f(x)>-infty } for every x ∈ X {displaystyle xin X} .