Equipartitions of measures in mathbb{R}⁴

We prove that each measure $\mu$ in $R^4$ admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional, affine subspace $L$ of $R^4$. Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on the Koschorke's exact singularity sequence and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4.