Convex homogeneous function

Question

Prove (or disprove) that any CONVEX function $f$, with the property that $\forall \alpha\ge 0, f(\alpha x) \le \alpha f(x)$, is positively homogeneous; i.e. $\forall \alpha\ge 0, f(\alpha x) = \alpha f(x)$.

Answer 1

Maybe I'm missing something, but it seems to me that you don't even need convexity. Given the property you stated, we have that, for $\alpha>0$, $$f(x)=f(\alpha^{-1}\alpha x)\leq \alpha^{-1}f(\alpha x)$$ so that $\alpha f(x)\leq f(\alpha x)$ as well. Therefore, we have that $\alpha f(x)=f(\alpha x)$ for every $\alpha>0$. The equality for $\alpha=0$ follows by continuity of $f$ at zero (which is implied by convexity).