In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to that convex combination of those domain elements. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called ****concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition

A real-valued function f on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any x and y in the interval and for any , A function is called strictly concave if for any and x \neq y. For a function, this second definition merely states that for every z strictly between x and y, the point (z, f(z)) on the graph of f is above the straight line joining the points (x, f(x)) and (y, f(y)). A function f is quasiconcave if the upper contour sets of the function are convex sets.

Properties

Functions of a single variable

f(x) = −x4 . C is concave if and only if it is midpoint concave, that is, for any x and y in C f(0) ≥ 0 , then f is subadditive on [0,\infty). Proof: 1 ≥ t ≥ 0 , letting y = 0 we have

Functions of n variables

Examples

Applications

Further References