|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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