Quasiconvexity

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 ${\displaystyle (-\infty ,a)}$ 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 ${\displaystyle f:S\to \mathbb {R} }$ defined on a convex subset S of a real vector space is quasiconvex if for all ${\displaystyle x,y\in S}$ and ${\displaystyle \lambda \in [0,1]}$ we have

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