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40th International Symposium on Computational Geometry, SoCG 2024

作者： Aronov, Boris Basit, Abdul Ramesh, Indu Tasinato, Gianluca Wagner, Uli Department of Computer Science and Engineering Tandon School of Engineering New York University BrooklynNY United States School of Mathematics Monash University Clayton Australia Institute of Science and Technology Austria Klosterneuburg Austria

ISBN: (纸本)9783959773164

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∗(n5/2). © Boris Aronov, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner.

关键词： Borsuk-Ulam Theorem Ham-Sandwich Theorem Mass partitions partitions of points in three dimensions

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Eight-Partitioning points in 3D, and Efficiently Too

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DISCRETE & COMPUTATIONAL GEOMETRY 2025年 1-25页

作者： Aronov, Boris Basit, Abdul Ramesh, Indu Tasinato, Gianluca Wagner, Uli NYU Tandon Sch Engn Dept Comp Sci & Engn Brooklyn NY 11201 USA Monash Univ Sch Math Clayton Vic 3800 Australia IST Austria Campus 1 A-3400 Klosterneuburg Austria

An eight-partition of a finite set of points (respectively, of a continuous mass distribution) in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>3$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>3$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>3$$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}<^>3$$\end{document} (with prescribed normal direction of one of the planes) in time O(n7/3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepac

关键词： Mass partitions partitions of points in three dimensions Borsuk-Ulam Theorem Ham-Sandwich Theorem

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