is defined by The effective domain of is equal to the image of 's epigraph under the canonical projection That is For a maximization problem (such as if the is concave rather than convex), the effective domain is instead equal to the image under of 's hypograph. Properties If a function takes the value such as if the function is real-valued, then its domain and effective domain are equal. A function is a proper convex function if and only if is convex, the effective domain of is nonempty, and for every See also References Convex analysis Functions and mappings