TY - GEN

T1 - Sylvester-Gallai for Arrangements of Subspaces

AU - Dvir, Zeev

AU - Hu, Guangda

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84958182842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958182842&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.SOCG.2015.29

DO - https://doi.org/10.4230/LIPIcs.SOCG.2015.29

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 29

EP - 43

BT - 31st International Symposium on Computational Geometry, SoCG 2015

A2 - Pach, Janos

A2 - Arge, Lars

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 31st International Symposium on Computational Geometry, SoCG 2015

Y2 - 22 June 2015 through 25 June 2015

ER -