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Eight-Partitioning Points in 3D, and Efficiently Too

Authors Boris Aronov , Abdul Basit , Indu Ramesh , Gianluca Tasinato , Uli Wagner

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Discrete mathematics
Keywords
  • Mass partitions
  • partitions of points in three dimensions
  • Borsuk-Ulam Theorem
  • Ham-Sandwich Theorem

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Abstract

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in ℝ³ consists of three planes that divide the space into 8 octants, such that each open octant contains at most 1/8 of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in ℝ³ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument.
We prove the following variant of this result: Any mass distribution (or point set) in ℝ³ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction.
Moreover, we present an efficient algorithm for calculating an eight-partition of a set of n points in ℝ³ (with prescribed normal direction of one of the planes) in time O^*(n^{5/2}).

Cite As Get BibTex

Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner. Eight-Partitioning Points in 3D, and Efficiently Too. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.8

Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Abdul Basit
  • School of Mathematics, Monash University, Clayton, Australia
Indu Ramesh
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Gianluca Tasinato
  • Institute of Science and Technology Austria, Klosterneuburg, Austria
Uli Wagner
  • Institute of Science and Technology Austria, Klosterneuburg, Austria

Funding

  • Aronov, Boris: Work has been supported by NSF grants CCF 15-40656 and CCF 20-08551, and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of this research was conducted while BA was visiting ISTA in the summers of 2022 and 2023. The visit of BA to ISTA in the summer of 2022 was supported by an ISTA Visiting Professorship.
  • Basit, Abdul: Work has been supported by Australian Research Council grant DP220102212.
  • Ramesh, Indu: Work supported by a Tandon School of Engineering Fellowship and by NSF Grant CCF-20-08551.

Acknowledgements

BA and AB would like to thank William Steiger for insightful initial discussions of the problems addressed in this work.

References

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