A counterexample to Gale's conjecture in ${\mathbb{R}}^3$

It is known that for every partition of the sphere $S^2$ of unit diameter into four parts, the largest diameter of a part is at least $\sqrt{(3+\sqrt{3})\mathrm{/}6}=0.88807\dots$, and this bound is attained. The conjecture of D.~Gale (1953) asserts that no set in ${\mathbb{R}}^3$ has a larger ratio. In this paper, we construct a $101$-point set $V\subset {\mathbb{R}}^3$ such that every partition of $V$ into four parts contains a part of diameter greater than $0.88928\mathrm{diam}V$.