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

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

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