quasikonvexen

Quasikonvexen, or quasiconvexity, is a concept in mathematical analysis and optimization describing a property of a real-valued function defined on a convex subset of a vector space. A function f: C → R, with C convex, is quasiconvex if for all x, y in C and all t in [0,1], f(tx+(1-t)y) ≤ max{ f(x), f(y) }. Equivalently, all sublevel sets { x in C : f(x) ≤ α } are convex for every α ∈ R.

In one dimension, quasiconvexity means the function is unimodal with a single global minimum along any line; in higher dimensions it means the function does not increase above the higher endpoint values when moving along a line segment.

Relation to convexity: Every convex function is quasiconvex, but the converse is false in general. Quasiconvexity is preserved under taking maximums: if f and g are quasiconvex, then h = max{f,g} is quasiconvex. If f is differentiable, a practical check is that all sublevel sets are convex; a simple example of a quasiconvex but not convex function on R is f(t) = arctan(t), which is increasing and has convex sublevel sets (-∞, tan α].

Applications: Quasikonvex functions arise in optimization, economics and decision theory because local minima are global on convex domains, and level-set structure guides efficient search methods. They enable useful theoretical guarantees even when full convexity does not hold.