Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a function differentiable on $\mathbb{R}^n$ and $\alpha$-convex, where $\alpha \in \mathbb{R}_+^{\star}$. Show that if $K$ is a non-empty closed convex set of $\mathbb{R}^n$, then $f$ admits a unique minimum on $K$.