Partition of three-dimensional sets into five parts of smaller diameter (Q1957037)

For integers \(n\), \(k\) and bounded sets \(C\) in \(\mathbb{R}^n\) define \(d^n_k(C)\) to be the smallest number \(d> 0\) such that \(C\) can be covered by \(k\) sets each with diameter not exceeding \(d\). A generalization of the classical Borsuk problem asks for \(d^n_k:=\sup\{dJ^n_k(C)\mid\text{diam\,}C= 1\}\). In this paper it is shown that \(d^5_3\leq 0,9425\), which improves the bound \(d^5_3\leq\sqrt{48^{-1}(35+ \sqrt{73})}= 0,9524\dots\) which is a special case of a result by M. Lassak for general \(n\). The proof uses an uncountable variety of universal cover systems consisting each of 3 sets.