Quasiconvex function

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 ( − ∞ , a ) {displaystyle (-infty ,a)} 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. 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 ( − ∞ , a ) {displaystyle (-infty ,a)} 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. A function f : S → R {displaystyle f:S o mathbb {R} } defined on a convex subset S of a real vector space is quasiconvex if for all x , y ∈ S {displaystyle x,yin S} and λ ∈ [ 0 , 1 ] {displaystyle lambda in } 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 f ( x ) {displaystyle f(x)} is to require that each sublevel set S α ( f ) = { x ∣ f ( x ) ≤ α } {displaystyle S_{alpha }(f)={xmid f(x)leq alpha }} is a convex set.