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Sylvester–Gallai for Arrangements of Subspaces

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

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dc.contributor.authorDvir, Zeev-
dc.contributor.authorHu, Guangda-
dc.date.accessioned2018-07-20T15:11:04Z-
dc.date.available2018-07-20T15:11:04Z-
dc.date.issued2016-04-08en_US
dc.identifier.citationDvir, Z, Hu, G. (2016). Sylvester–Gallai for Arrangements of Subspaces. Discrete and Computational Geometry, 56 (940 - 965. doi:10.1007/s00454-016-9781-7en_US
dc.identifier.urihttp://arks.princeton.edu/ark:/88435/pr1m977-
dc.description.abstractIn this work we study arrangements of k-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 2k-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. (Proc Natl Acad Sci USA 110(48):19213–19219, 2013). One of the main ingredients in the proof is a strengthening of a theorem of Barthe (Invent Math 134(2):335–361, 1998) (from the k= 1 to k> 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).en_US
dc.format.extent940 - 965en_US
dc.language.isoen_USen_US
dc.relation.ispartofDiscrete and Computational Geometryen_US
dc.rightsAuthor's manuscripten_US
dc.titleSylvester–Gallai for Arrangements of Subspacesen_US
dc.typeJournal Articleen_US
dc.identifier.doidoi:10.1007/s00454-016-9781-7-
dc.date.eissued2016en_US
pu.type.symplectichttp://www.symplectic.co.uk/publications/atom-terms/1.0/journal-articleen_US

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