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.
Read more about this topic: Quasiconvex Function
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