Equipartitions of measures in $\mathbb{R}^4$

A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) $d\mu = f\, dm$ on $\mathbb{R}^n$ there exist $n$ hyperplanes dividing $\mathbb{R}^n$ into $2^n$ parts of equal measure. It is known that the answer is positive in dimension $n=3$ (see H. Hadwiger (1966)) and negative for $n\geq 5$ (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grünbaum's problem in the critical dimension $n=4$ by proving that each measure $\mu$ in $\mathbb{R}^4$ admits an equipartition by $4$ hyperplanes, provided that it is symmetric with respect to a $2$-dimensional affine subspace $L$ of $\mathbb{R}^4$. 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 Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension $4$; see G. C. Tootill (1956) and D. E. Knuth (2001).