Quasiconvex Function - Definition and Properties

Definition and Properties

A function defined on a convex subset S of a real vector space is quasiconvex if for all and we have

In words, if f is such that it is always true that a point directly between two other points does not give a higher a value of the function than do both of the other points, then f is quasiconvex. Note that the points x and y, and the point directly between them, can be points on a line or more generally points in n-dimensional space.

An alternative way (see introduction) of defining a quasi-convex function is to require that each sub-levelset is a convex set.

If furthermore

for all and, then is strictly quasiconvex. That is, strict quasiconvexity requires that a point directly between two other points must give a lower value of the function than one of the other points does.

A quasiconcave function is a function whose negative is quasiconvex, and a strictly quasiconcave function is a function whose negative is strictly quasiconvex. Equivalently a function is quasiconcave if

and strictly quasiconcave if

A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour sets.

A function that is both quasiconvex and quasiconcave is quasilinear.