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,, 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 onto X. That is
Note that if a convex function is mapping to the normal real number line given by then the effective domain is the same as the normal definition of the domain.
A function is a proper convex function if and only if f is convex, the effective domain of f is nonempty and for every.
