Geodesic
Partition of Three-Dimensional Sets into Five Parts of Smaller Diameter
Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 233-245

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The classical Borsuk problem on partitioning sets into pieces of smaller diameter is considered. A new upper bound for $$ d_5^3=\sup_{\Phi\subset\mathbb R^3,\operatorname{diam}\Phi=1}\inf\{x\ge0:\Phi=\Phi_1\cup\Phi_2\cup\dots\cup\Phi_5,\operatorname{diam}\Phi_i\le x\} $$ is given, which improves the previous bound obtained by Lassak in 1982.
Keywords: Borsuk's problem, diameter of a set, Lassak's bound, Jung's ball, Helly's theorem, isometry.
Mots-clés : partition of 3D sets, Gale's conjecture 
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     title = {Partition of {Three-Dimensional} {Sets} into {Five} {Parts} of {Smaller} {Diameter}},
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A. B. Kupavskii; A. M. Raigorodskii. Partition of Three-Dimensional Sets into Five Parts of Smaller Diameter. Matematičeskie zametki, Tome 87 (2010) no. 2, pp. 233-245. http://geodesic.mathdoc.fr/item/MZM_2010_87_2_a5/