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 → R ∪ { ± ∞ } , 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 ∪ { ± ∞ } onto X. That is

Note that if a convex function is mapping to the normal real number line given by f : X → R then the effective domain is the same as the normal definition of the domain.

A function f : X → R ∪ { ± ∞ } is a proper convex function if and only if f is convex, the effective domain of f is nonempty and f ( x ) > − ∞ for every x ∈ X .