Weight Balancing on Boundaries

Barba, L. and Cheong, O. and Dobbins, M.G. and Fleischer, R. and Kawamura, A. and Pach, János (2022) Weight Balancing on Boundaries. JOURNAL OF COMPUTATIONAL GEOMETRY, 13 (1). pp. 1-12. ISSN 1920-180X


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Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any d-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a ⌊d/2⌋-face and a point on a ⌈d/2⌉-face. © 2022, Carleton University. All rights reserved.

Item Type: Article
Additional Information: Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland School of Computing, KAIST, Daejeon, South Korea Department of Mathematics, Binghamton University, Binghamton, NY, United States Department of Computer Science, Heinrich Heine University Düsseldorf, Germany Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan Siemens Electronic Design Automation, United States Department of Computer and Network Engineering, University of Electro-Communications, Tokyo, Japan Rényi Institute, Budapest, Hungary Moscow Institute of Physics and Technology, Moscow, Russian Federation Software School, Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai, China School of Engineering, Kwansei Gakuen University, Japan Shopify, Ottawa, ON, Canada School of Computer Science, Carleton University, Ottawa, ON, Canada Export Date: 18 January 2023 Funding details: Natural Sciences and Engineering Research Council of Canada, NSERC Funding details: European Research Council, ERC, 882971 Funding details: University of Sydney, Usyd Funding details: National Natural Science Foundation of China, NSFC, 60973026 Funding details: Austrian Science Fund, FWF, Z 342-N31 Funding details: Science and Technology Commission of Shanghai Municipality, STCSM, 08DZ2271800, 09DZ2272800 Funding details: Ministry of Education and Science of the Russian Federation, Minobrnauka, 075-15-2019-1926 Funding details: National Research Foundation of Korea, NRF, 2011-0030044 Funding details: The Research Council, TRC Funding details: Norges Forskningsråd Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, KKP-133864 Funding details: Shanghai Leading Academic Discipline Project, B114 Funding details: National Research, Development and Innovation Office Funding text 1: This work was initiated at the Fields Workshop on Discrete and Computational Geometry (15th Korean Workshop on Computational Geometry), Ottawa, Canada, in August 2012, and it was continued at IPDF 2012 Workshop on Algorithmics on Massive Data, held in Sunshine Coast, Australia, August 23–27, 2012, supported by the University of Sydney International Program Development Funding, a workshop in Ikaho, Japan, and 16th Korean Workshop on Computation Geometry in Kanazawa, Japan. The authors are grateful to the organizers of these workshops. They also thank John Iacono and Pat Morin for pointing out Proposition 1, as well as Jean-Lou De Carufel and Tianhao Wang for helpful discussions. Funding text 2: This work was supported by NSF China (no. 60973026); the Shanghai Leading Academic Discipline Project (no. B114); the Shanghai Committee of Science and Technology (nos. 08DZ2271800 and 09DZ2272800); the Research Council (TRC) of the Sultanate of Oman; NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea; the National Research, Development and Innovation Office, NKFIH, project KKP-133864; the Austrian Science Fund (FWF), grant Z 342-N31; ERC Advanced Grant "GeoScape" (no. 882971); the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant (no. 075-15-2019-1926); and the Natural Sciences and Engineering Research Council of Canada (NSERC).
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria
Depositing User: MTMT SWORD
Date Deposited: 30 Mar 2023 13:22
Last Modified: 30 Mar 2023 13:22

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