Partition of three-dimensional sets into five parts of smaller diameter
Andrey Borisovich Kupavskii, Andrei Mikhailovich Raigorodskii · Mathematical Notes · 2010
The classical Borsuk problem on partitioning sets into pieces of smaller diameter is considered. A new upper bound for $$ d_5^3 = \mathop {\sup }\limits_{\Phi \subset \mathbb{R}^3 ,diam \Phi = 1} \inf \{ x \geqslant 0:\Phi = \Phi _1 \cup \Phi _2 \cup \cdots \cup \Phi _5 ,diam \Phi _i \leqslant x\} $$ is given, which improves the previous bound obtained by Lassak in 1982.