Equipartitions of measures in ℝ4

Abstract

A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) dμ = f dm on ℝn there exist n hyperplanes dividing ℝn into 2n 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 Grünbaum's problem in the critical dimension n = 4 by proving that each measure μin ℝ4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional affine subspace L of ℝ4. 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). © 2007 American Mathematical Society.