Let $f$ be a function which all stationary points are global minima. This type of functions are also known as invex. It means, there exits $\eta(x,y)$ such that $f(x)-f(y) \geq \zeta_{y}^{T}\eta(x,y)$ for all $x,y \in \mathbb{R}^{n}$, and $\zeta_{y}\in \partial f(y)$ (in the Clarke sense). Considering this, is it possible to prove or disprove that $f(x) + \frac{\alpha}{2}\lVert x \rVert^{2}$ is convex for some $\alpha>0$?
I have checked numerically several examples and it seems to be true, however I cannot see how to prove this (or disprove it).