Definition. A set is said to be concave if for every vector and in , is in for all . In other words, all points on the line between two points of lie in .
Definition. A functional is a mapping from a vector space to the real numbers .
Thus, for example, every norm is a functional.
Definition. A linear functional is a functional such that for each and in the linear space , and for all scalars and , we have .
Definition. The norm of a linear functional is defined on the normed linear space by
Definition. A functional defined on a concave subset of a vector space is said to be concave on if for every vector and in ,
Definition. A local minimizer of a real-valued function is any such that in some neighborhood of .
Definition. A global minimizer of a real-valued function on a set is any such that for all .
Definition. A cluster point of a sequence is any point such that every neighborhood of contains at least one .
Definition. The concave hull of a set in a metric space is the smallest concave set containing .