Linear Perturbations of Quasiconvex Functions and Convexity

Let $E$ be a real vector space with dual space $E^{\ast }$ and let $C\subset E$ be a convex subset with more than one point. Let $f:C\to \mathbb{R}$ be a function satisfying a mild stability property at ‘flat’ points of the (relative) boundary of $C$. We show that $f$ is convex if and only if for some linear form $c^{\ast }$ on $E$ not constant on $C$, the function $f+\lambda c^{\ast }$ is quasiconvex for all $\lambda \in \mathbb{R}$.