Smooth convex extensions of convex functions

Let $C$ be a compact convex subset of $\mathbb{R}^n$, $f:C\to\mathbb{R}$ be a convex function, and $m\in\{1, 2, ..., \infty\}$. Assume that, along with $f$, we are given a family of polynomials satisfying Whitney's extension condition for $C^m$, and thus that there exists $F\in C^{m}(\mathbb{R}^n)$ such that $F=f$ on $C$. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that $F$ can be taken to be convex as well. We give a satisfactory solution to this problem in the case $m=\infty$, and also less satisfactory solutions in the case of finite $m\geq 2$ (nonetheless obtaining an almost optimal result for $C$ a finite intersection of ovaloids). For a solution to a similar problem in the case $m=1$ (even for $C$ not necessarily convex), see arXiv:1507.03931, arXiv:1706.09808, arXiv:1706.02235.