page_title: Quasiconvex function
section_title: Definition and properties
hierarchical_section_title: Quasiconvex function / Definition and properties
caption_attribution_description: Illustration of en:Monotonic function
context_page_description: In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave.
  All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Quasiconvexity and quasiconcavity extend to functions with multiple arguments the notion of unimodality of functions with a single real argument.
context_section_description: 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 value of the function than both of the other points do, 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 sublevel set
  
  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.
  A particular case of quasi-concavity, if , is unimodality, in which there is a locally maximal value.
caption: A quasilinear function is both quasiconvex and quasiconcave.
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