3 research outputs found
On the multiple Borsuk numbers of sets
The 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
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.Comment: 16 pages, 3 figure
On the multiple Borsuk numbers of sets
The \emph{Borsuk number} of a set of diameter in Euclidean -space is the smallest value of such that can be partitioned into sets of diameters less than . Our aim is to generalize this notion in the following way: The \emph{-fold Borsuk number} of such a set is the smallest value of such that there is a -fold cover of with sets of diameters less than . In this paper we characterize the -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 -space