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

is a convex set .

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., $f(\lambda x+(1-\lambda)y)\le \alpha$.