Universal Taylor Series On Convex Subsets Of $\text{Mathbb}{C}^N$

We prove the existence of holomorphic functions $f$ defined on any open convex subset $\mathrm{\Omega }\subset {\mathbb{C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from $\bar{\mathrm{\Omega }}$ and on denumerably many convex compact sets in ${\mathbb{C}}^n\mathrm{\backslash }\mathrm{\Omega }$ which may meet the boundary $\mathrm{\partial }\mathrm{\Omega }$. If the universal approximation is only required on convex compact sets disjoint from $\bar{\mathrm{\Omega }}$, then $f$ may be chosen to be smooth on $\mathrm{\partial }\mathrm{\Omega }$, that is $f\in A^{\infty}({\rm \Omega})$. Those are generic universalities.