Partition of Three-Dimensional Sets into Five Parts of Smaller Diameter

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.