Equipartitions of measures in R4

A well-known problem of B. Grunbaum (1960) asks whether for every continuous mass distribution (measure) d mu = f dm on R-n there exist n hyperplanes dividing 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 >= 5 (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grunbaum's problem in the critical dimension n = 4 by proving 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 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).