Abstract
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.
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Dedicated to Aladár Heppes on his 80th birthday
The support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences is gratefully acknowledged
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Hujter, M., Lángi, Z. On the multiple Borsuk numbers of sets. Isr. J. Math. 199, 219–239 (2014). https://doi.org/10.1007/s11856-013-0048-1
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DOI: https://doi.org/10.1007/s11856-013-0048-1