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 .
Thus, for example, every norm is a functional.
Definition. The norm of a linear functional is defined on the normed linear space by
A concave functional has the property that its values along a line segment lie below or on the line between its values at the end points. The functional is strictly concave on if strict inequality holds above for . Finally, isuniformly concave on if there exists such that for all ,