Equipartitions of measures in $\mathbb {R}^4$
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- by Rade T. Živaljević PDF
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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).References
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Additional Information
- Rade T. Živaljević
- Affiliation: Mathematical Institute SANU, Knez Mihailova 35/1, P.O. Box 367, 11001 Belgrade, Serbia
- Email: rade@turing.mi.sanu.ac.yu
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 153-169
- MSC (2000): Primary 54C40, 14E20; Secondary 46E25, 20C20
- DOI: https://doi.org/10.1090/S0002-9947-07-04294-8
- MathSciNet review: 2341998