REAL

# On the multiple Borsuk numbers of sets

Hujter, Mihály and Lángi, Zsolt (2014) On the multiple Borsuk numbers of sets. ISRAEL JOURNAL OF MATHEMATICS, 199. pp. 219-239. ISSN 0021-2172

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The \emph{Borsuk number} of a set $S$ of diameter $d >0$ in Euclidean $n$-space is the smallest value of $m$ such that $S$ can be partitioned into $m$ sets of diameters less than $d$. Our aim is to generalize this notion in the following way: The \emph{$k$-fold Borsuk number} of such a set $S$ is the smallest value of $m$ such that there is a $k$-fold cover of $S$ with $m$ sets of diameters less than $d$. In this paper we characterize the $k$-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean $3$-space.