Sylvester–Gallai for Arrangements of Subspaces

Abstract

In this work we study arrangements of k-dimensional subspaces V 1 , . . . , V n ⊂ C ℓ . 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 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 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. [BDWY13].One of the main ingredients in the proof is a strengthening of a Theorem of Barthe [Bar98] (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).

Rank bounds for design matrices with block entries and geometric applications

“…The second application is a generalization of the Sylvester-Gallai theorem for arrangements of subspaces. Such a result was recently proved in [DH16] and we are able to give an asymptotically tight improvement to their results. The last application involves arrangements of lines and curves in C d that have many pairwise incidences (each line/curve intersects many others).…”

“…The original bound proven in [DH16] was a slightly worse O( 4 /δ 2 ). For δ = 1 and = 1 the bound of 3 we get is completely tight as there are three dimensional configurations of one dimensional subspaces over C with every pair spanning some third subspace (this can be obtained by taking the Hesse configuration and moving to projective space [AD09]).…”

“…The condition V i ∩ V i = { 0} is needed due to the following example given in [DH16]: Set = 2 and n = d(d −1)/2 and let { e 1 , e 2 , . .…”

A Forward-Reverse Brascamp-Lieb Inequality: Entropic Duality and Gaussian Optimality

“…Furthermore, the Prékopa-Leindler inequality can be seen as a special case of Barthe's inequality. Due in part to their utility in establishing impossibility bounds, these functional inequalities have attracted much attention in information theory [10][11][12][13][14][15][16][17], theoretical computer science [18][19][20][21][22] and statistics [23][24][25][26][27][28], to name only a small subset of the literature. Over the years, various proofs of these inequalities have been proposed [1,[29][30][31][32][33][34].…”

Vertex numbers of simplicial complexes with free abelian fundamental group

“…The bound |R| ≥ n 2 already gives a Ω(n 2/3 ) lower bound for vertex numbers in Theorem 1.1(b). To strengthen this bound, we use extremal results in discrete geometry (specifically, the fractional Sylvester-Gallai results in [6,11,10]), to show that any 3-presentation S|R ∼ = Z n has |S| = Ω(n 3/2 ). This translates to a Ω(n 3/4 ) lower bound in Theorem 1.1(b).…”