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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 uniformlycontinuous function, with modulus of continuity $\omega$. We provide anecessary and sufficient condition on $f$, $G$ for the existence of a convexfunction $F\in C^{1, \omega}(\mathbb{R}^n)$ such that $F=f$ on $C$ and $\nablaF=G$ on $C$, with a good control of the modulus of continuity of $\nabla F$ interms of that of $G$. On the other hand, assuming that $C$ is compact, we alsosolve a similar problem for the class of $C^1$ convex functions on$\mathbb{R}^n$, with a good control of the Lipschitz constants of theextensions (namely, $\textrm{Lip}(F)\lesssim \|G\|_{\infty}$). Finally, we givea geometrical application concerning interpolation of compact subsets $K$ of$\mathbb{R}^n$ by boundaries of $C^1$ or $C^{1,1}$ convex bodies withprescribed outer normals on $K$.

Whitney Extension Theorems for convex functions of the classes $C^1$ and $C^{1,\omega}$