quasiconvexity

Quasiconvexity is a property of a real-valued function defined on a convex subset of a vector space that generalizes convexity. A function f: D → R with D convex is quasiconvex if all its sublevel sets Sα = {x ∈ D : f(x) ≤ α} are convex for every real α.

Equivalently, for all x, y ∈ D and all t ∈ [0,1], f(tx + (1 − t)y) ≤ max{f(x), f(y)}. This line-segment condition provides a practical way to verify quasiconvexity in many settings.

Every convex function is quasiconvex, but the converse is false in general. A simple example that is quasiconvex but not convex is the step function f(x) = 0 for x ≤ 0 and f(x) = 1 for x > 0; its sublevel sets are (-∞, 0] and the whole real line, both convex.

Strict quasiconvexity strengthens this by requiring strict convexity of sublevel sets. Equivalently, if f(x) ≠ f(y), then f(tx + (1 − t)y) < max{f(x), f(y)} for all t ∈ (0,1).

Quasiconvex functions enjoy several useful properties: the pointwise maximum of quasiconvex functions is quasiconvex; the minimum need not be. On a convex domain, every local minimum of a quasiconvex function is a global minimum. In optimization and economics, quasiconvexity often suffices to ensure tractable search for minima even when full convexity is absent.