Knowlpedia

Quasiconvex function

A function whose value on a line segment never exceeds the larger endpoint value.

Let XX be a real and let f:X→(−∞,∞]f:X\to(-\infty,\infty] be an extended-real-valued function. The function ff is quasiconvex if, for all x,y∈Xx,y\in X and λ∈[0,1]\lambda\in[0,1],

f(λx+(1−λ)y)≤max⁡{f(x),f(y)}.f(\lambda x+(1-\lambda)y)\le \max\{f(x),f(y)\}.

Equivalently, every sublevel set {x∈X:f(x)≤α}\{x\in X:f(x)\le \alpha\} is convex.

Examples
  • Any is quasiconvex.
  • The function f(x)=∣x∣f(x)=\sqrt{|x|} on R\mathbb R is quasiconvex but not convex.
  • Any constant function is quasiconvex.
Remarks

Quasiconvexity is weaker than convexity: it controls sublevel sets but does not require the graph to lie below its chords.