On multiple Borsuk numbers in normed spaces

Hujter and L'angi introduced the $k$-fold Borsuk number of a set $S$ in Euclidean $n$-space of diameter $d>0$ as the smallest cardinality of a family $\mathcal{F}$ of subsets of $S$, of diameters strictly less than $d$, such that every point of $S$ belongs to at least $k$ members of $\mathcal{F}$. We investigate whether a $k$-fold Borsuk covering of a set $S$ in a finite dimensional real normed space can be extended to a completion of $S$. Furthermore, we determine the $k$-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.