### Abstract

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, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) 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 bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

Original language | English |
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Title of host publication | Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014 |

Publisher | Association for Computing Machinery |

Pages | 436-443 |

Number of pages | 8 |

ISBN (Print) | 9781450325943 |

DOIs | |

Publication status | Published - 2014 |

Event | 30th Annual Symposium on Computational Geometry, SoCG 2014 - Kyoto, Japan Duration: Jun 8 2014 → Jun 11 2014 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Other

Other | 30th Annual Symposium on Computational Geometry, SoCG 2014 |
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Country | Japan |

City | Kyoto |

Period | 6/8/14 → 6/11/14 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

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## Cite this

*Proceedings of the 30th Annual Symposium on Computational Geometry, SoCG 2014*(pp. 436-443). (Proceedings of the Annual Symposium on Computational Geometry). Association for Computing Machinery. https://doi.org/10.1145/2582112.2582142