Authors: | Živaljević, Rade | Affiliations: | Mathematical Institute of the Serbian Academy of Sciences and Arts | Title: | Equipartitions of measures in ℝ4 | Journal: | Transactions of the American Mathematical Society | Volume: | 360 | Issue: | 1 | First page: | 153 | Last page: | 169 | Issue Date: | 1-Jan-2008 | Rank: | M21 | ISSN: | 0002-9947 | DOI: | 10.1090/S0002-9947-07-04294-8 | 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. |
Keywords: | Geometric combinatorics | Gray codes | Partitions of masses | Publisher: | American Mathematical Society | Project: | Serbian Ministry of Science and Technology, Grant no. 1643 |
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