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ON THE MULTIPLE BORSUK NUMBERS OF SETS
"... Abstract. The Borsuk number of a set S of diameter d> 0 in Euclidean nspace 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 kfold Borsuk number of such a set S is the smallest value of ..."
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Abstract. The Borsuk number of a set S of diameter d> 0 in Euclidean nspace 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 kfold Borsuk number of such a set S is the smallest value of m such that there is a kfold cover of S with m sets of diameters less than d. In this paper we characterize the kfold 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 3space. 1.