  Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex),
$f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly
continuous function, with modulus of continuity $\omega$. We provide a
necessary and sufficient condition on $f$, $G$ for the existence of a convex
function $F\in C^{1, \omega}(\mathbb{R}^n)$ such that $F=f$ on $C$ and $\nabla
F=G$ on $C$, with a good control of the modulus of continuity of $\nabla F$ in
terms of that of $G$. On the other hand, assuming that $C$ is compact, we also
solve a similar problem for the class of $C^1$ convex functions on
$\mathbb{R}^n$, with a good control of the Lipschitz constants of the
extensions (namely, $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$). Finally, we give
a geometrical application concerning interpolation of compact subsets $K$ of
$\mathbb{R}^n$ by boundaries of $C^1$ or $C^{1,1}$ convex bodies with
prescribed outer normals on $K$.
