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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Equipartitions of measures in $ \mathbb{R}^4$


Author: Rade T. Zivaljevic
Journal: Trans. Amer. Math. Soc. 360 (2008), 153-169
MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
Published electronically: June 27, 2007
MathSciNet review: 2341998
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Abstract: A well-known problem of B. Grünbaum (1960) 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 (1966)) and negative for $ n\geq 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 $ \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).


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Additional Information

Rade T. Zivaljevic
Affiliation: Mathematical Institute SANU, Knez Mihailova 35/1, P.O. Box 367, 11001 Belgrade, Serbia
Email: rade@turing.mi.sanu.ac.yu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04294-8
PII: S 0002-9947(07)04294-8
Keywords: Geometric combinatorics, partitions of masses, Gray codes.
Received by editor(s): February 28, 2005
Received by editor(s) in revised form: September 14, 2005
Published electronically: June 27, 2007
Additional Notes: The author was supported by the grant 1643 of the Serbian Ministry of Science and Technology.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.