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Transactions of the American Mathematical Society

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Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem


Authors: Pavle V. M. Blagojević, Florian Frick, Albert Haase and Günter M. Ziegler
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
Published electronically: July 5, 2018
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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
Email: ff238@cornell.edu

Albert Haase
Affiliation: Institute of Mathematics, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
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

DOI: https://doi.org/10.1090/tran/7528
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”.
Article copyright: © Copyright 2018 American Mathematical Society

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