Let $\varphi$ be a nonnegative function defined on a domain $\omega\subset R^n$ such that $\varphi$ is $C^2$ and convex on $\omega$. Does $\varphi$ admit a $C^2-$ extension on $\Omega\supset \overline{\omega}$ which is nonnegative and convex on $\Omega?$
For $n=1$ and $\omega =(a,b)$, a possible extension on $(b,+) $ is $\overline{\varphi}(x)=\varphi'(b)(x-b)+\varphi(b)$. A second idea consists on using odd polynomial, but this does not satisfy the conditions we ask for, unless additional assumtions are imposed. Also I can't see how to extend this construction to $n\ge 2. $