quasikonvexe

Quasikonvexe, commonly called quasi-convex in English, refers to a property of functions whose lower level sets are convex. Let f: D → R be defined on a convex subset D of R^n. The function is quasi-convex if for all x and y in D and all t in [0,1], f(t x + (1−t) y) ≤ max{f(x), f(y)}. Equivalently, for every α ∈ R the sublevel set {x ∈ D | f(x) ≤ α} is convex. This perspective emphasizes the shape of the level contours rather than the graph of the function.

Quasi-convexity is weaker than convexity: every convex function is quasi-convex, but there exist non-convex quasi-convex functions. Examples include the indicator function of a convex set, I_C(x) = 0 if x ∈ C and ∞ otherwise, which is quasi-convex; the function f(x) = max{a_i^T x + b_i} over a finite family of affine functions is quasi-convex because it is the maximum of affine (hence convex) functions. More generally, a nondecreasing transformation of a quasi-convex function preserves quasi-convexity in many cases.

If f is differentiable, a standard first-order condition holds: f is quasi-convex if and only if f(y) ≤ f(x) implies ∇f(x) · (y − x) ≤ 0 for all x, y in D. This provides a practical check via first-order information on smooth functions.

In optimization on convex domains, any local minimum of a quasi-convex function is a global minimum, making local search methods effective under suitable regularity. Quasikonvex functions arise in economics, statistics, and optimization, where weak monotone relationships or single-peaked preferences can be modeled without full convexity.