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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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}).

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)

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@InProceedings{aronov_et_al:LIPIcs.SoCG.2024.8, author = {Aronov, Boris and Basit, Abdul and Ramesh, Indu and Tasinato, Gianluca and Wagner, Uli}, title = {{Eight-Partitioning Points in 3D, and Efficiently Too}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.8}, URN = {urn:nbn:de:0030-drops-199538}, doi = {10.4230/LIPIcs.SoCG.2024.8}, annote = {Keywords: Mass partitions, partitions of points in three dimensions, Borsuk-Ulam Theorem, Ham-Sandwich Theorem} }

Document

**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.

Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf. On the Number of Ordinary Lines Determined by Sets in Complex Space. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{basit_et_al:LIPIcs.SoCG.2017.15, author = {Basit, Abdul and Dvir, Zeev and Saraf, Shubhangi and Wolf, Charles}, title = {{On the Number of Ordinary Lines Determined by Sets in Complex Space}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.15}, URN = {urn:nbn:de:0030-drops-71883}, doi = {10.4230/LIPIcs.SoCG.2017.15}, annote = {Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method} }

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