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

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 $ω$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\in C^{1, ω}(\mathbb{R}^n)$ 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 $K$.

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