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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Topology of the Grünbaum–Hadwiger–Ramos hyperplane mass partition problem
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by Pavle V. M. Blagojević, Florian Frick, Albert Haase and Günter M. Ziegler PDF
Trans. Amer. Math. Soc. 370 (2018), 6795-6824 Request permission

Abstract:

In 1960 Grünbaum asked whether for any finite mass in $\mathbb {R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le 3$, but disproved by Avis (1984) for $d\ge 5$, while the case $d=4$ remained open.

More generally, Ramos (1996) asked for the smallest dimension $\Delta (j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $\Delta (j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica and Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology.

We give a critical review of the work on the Grünbaum–Hadwiger–Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that $\Delta (j,2)= \frac 12(3j+1)$ in the cases when $j-1$ is a power of $2$, $j\ge 5$.

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Additional Information
  • Pavle V. M. Blagojević
  • Affiliation: Institute of Mathematics, FU Berlin, Arnimallee 2, 14195 Berlin, Germany—and—Matematički Institut SANU, Knez Mihailova 36, 11001 Beograd, Serbia
  • Email: blagojevic@math.fu-berlin.de
  • Florian Frick
  • Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
  • MR Author ID: 1079440
  • ORCID: 0000-0002-7635-744X
  • Email: ff238@cornell.edu
  • Albert Haase
  • Affiliation: Institute of Mathematics, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
  • MR Author ID: 1178068
  • Email: albert.haase@gmail.com
  • Günter M. Ziegler
  • Affiliation: Institute of Mathematics, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
  • Email: ziegler@math.fu-berlin.de
  • Received by editor(s): February 12, 2015
  • Published electronically: July 5, 2018
  • Additional Notes: The research by Pavle V. M. Blagojević leading to these results has received funding from the Leibniz Award of Wolfgang Lück granted by DFG. This work was also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.
    The research of Florian Frick and of Albert Haase leading to these results has received funding from German Science Foundation DFG via the Berlin Mathematical School.
    The research by Günter M. Ziegler leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels and from the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 6795-6824
  • MSC (2010): Primary 52A35, 55N25; Secondary 51M20, 55R20
  • DOI: https://doi.org/10.1090/tran/7528
  • MathSciNet review: 3841833