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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Equipartitions of measures in $\mathbb {R}^4$
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by Rade T. Živaljević PDF
Trans. Amer. Math. Soc. 360 (2008), 153-169 Request permission

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. 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