Quasiconvexity via convex sublevel sets

Proposition. Let $X$ be a vector space. A function $f:X\to\mathbb{R}$ is quasiconvex if and only if for every $\alpha\in\mathbb{R}$ the sublevel set

L_\alpha:=\{x\in X: f(x)\le \alpha\}

is a convex set.

Remarks

Context. This is the defining geometric feature of quasiconvexity: convexity is demanded of level sets rather than of the epigraph.

Proof sketch. (⇒) If $x,y\in L_\alpha$ , then $\max\{f(x),f(y)\}\le \alpha$ , hence $f(\lambda x+(1-\lambda)y)\le \alpha$ , so the combination lies in $L_\alpha$ . (⇐) Fix $x,y$ and let $\alpha=\max\{f(x),f(y)\}$ . Then $x,y\in L_\alpha$ , so convexity of $L_\alpha$ gives $\lambda x+(1-\lambda)y\in L_\alpha$ , i.e., $f(\lambda x+(1-\lambda)y)\le \alpha$ .