Equipartitions of measures in \(\mathbb{R}^4\). (English) Zbl 1202.68454
Summary: A well-known problem of B. Grünbaum [“Partitions of mass-distributions and of convex bodies by hyperplanes,” Pac. J. Math. 10, 1257–1261 (1960; Zbl 0101.14603)] 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 [“Simultane Vierteilung zweier Körper,”Arch. Math. 17, 274–278 (1966; Zbl 0137.41501)]) and negative for \( n\geq 5\) (see D. Avis [“Non-partitionable point sets,” Inf. Process. Lett. 19, 125–129 (1985; Zbl 0564.51007)] and E. Ramos [“Equipartition of mass distributions by hyperplanes,” Discrete Comput. Geom. 15, No.2, 147–167(1996; Zbl 0843.68120)]). 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).
MSC:
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U10 | Computing methodologies for image processing |
| 14E20 | Coverings in algebraic geometry |
| 46E25 | Rings and algebras of continuous, differentiable or analytic functions |
| 20C20 | Modular representations and characters |
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