On the Borsuk number of four-dimensional sets

Details On the Borsuk number of four-dimensional sets On the Borsuk number of four-dimensional sets

2010-07-15

arxiv.org/abs/1007.2518v2

Mathematics

Metric Geometry

This paper has been withdrawn by the author due to a crucial computational error in the proof of the lemma

Scientific paper

Borsuk conjectured that every n-dimensional bounded set of positive diameter can be partitioned into n+1 sets of smaller diameters. This conjecture was proved for n=2 by Borsuk, for n=3 first by Eggleston, and disproved for n > 297 by Hinrichs and Richer. It is not known if the conjecture holds for 3 < n < 298. The best upper bound for the number of subsets of smaller diameters a four-dimensional set can be partitioned into is nine. This estimate was given by Lassak in 1982. In this note we improve this estimate by one.

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