Range of the gradient

Question

Let $f \in C^2(\Omega \subseteq \Bbb{R}^n)$ be a real valued and strictly convex (meaning that $(\partial_{ij}f(\underline{x}))$ is positive definite) function on a convex, open, connected subset $\Omega$ of $\Bbb R^n$. I would like to know wether there is some constraint that can be derived on the range of the gradient of $f$. The motivation behind this question is that I would like to characterize the domain of definition of the Legendre Transform of $f$.