### Abstract

In this work we study arrangements of κ-dimensional subspaces V_{1}, . . . , V_{n} ⊂ Cl. Our main result shows that, if every pair V_{a}, V_{b} of subspaces is contained in a dependent triple (a triple V_{a}, V_{b}, V_{c} contained in a 2κ-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that V_{a} \ V_{b} = {0} for every pair (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly s theorem for complex numbers), which proves the k = 1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. [1]. One of the main ingredients in the proof is a strengthening of a theorem of Barthe [3] (from the k = 1 to κ>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).

Original language | English (US) |
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Title of host publication | 31st International Symposium on Computational Geometry, SoCG 2015 |

Editors | Janos Pach, Lars Arge |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 29-43 |

Number of pages | 15 |

ISBN (Electronic) | 9783939897835 |

DOIs | |

State | Published - Jun 1 2015 |

Event | 31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands Duration: Jun 22 2015 → Jun 25 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 34 |

### Other

Other | 31st International Symposium on Computational Geometry, SoCG 2015 |
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Country | Netherlands |

City | Eindhoven |

Period | 6/22/15 → 6/25/15 |

### All Science Journal Classification (ASJC) codes

- Software

## Cite this

*31st International Symposium on Computational Geometry, SoCG 2015*(pp. 29-43). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 34). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SOCG.2015.29