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