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Sylvester-Gallai for Arrangements of Subspaces

Author(s): Dvir, Zeev; Hu, G

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Abstract: In this work we study arrangements of κ-dimensional subspaces V1, . . . , Vn ⊂ Cl. Our main result shows that, if every pair Va, Vb of subspaces is contained in a dependent triple (a triple Va, Vb, Vc 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 Va \ Vb = {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).
Publication Date: 2015
Electronic Publication Date: 2015
Citation: Dvir, Z, Hu, G. (2015). Sylvester-Gallai for Arrangements of Subspaces. 34 (29 - 43. doi:10.4230/LIPIcs.SOCG.2015.29
DOI: doi:10.4230/LIPIcs.SOCG.2015.29
Pages: 29 - 43
Type of Material: Conference Article
Journal/Proceeding Title: 31st International Symposium on Computational Geometry, SoCG 2015
Version: Author's manuscript

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