TY - GEN

T1 - Overlap properties of geometric expanders

AU - Fox, Jacob

AU - Gromov, Mikhail

AU - Lafforgue, Vincent

AU - Naor, Assaf

AU - Pach, János

PY - 2011

Y1 - 2011

N2 - The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ∈ (0,1] such that no matter how we map the vertices of H into ℝd there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. In [18], motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {Hn}n=1∞ of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which infn≥1 c(Hn) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any d and any ε > 0, there exists K = K(ε,d) ≥ d + 1 satisfying the following condition. For any k ≥ K, for any point q ∈ ℝd and for any finite Borei measure μ on ℝd with respect to which every hyperplane has measure 0, there is a partition ℝd = A1 ∪. .. ∪ Ak into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples Ai1,..., A id+1 have the property that either all simplices with one vertex in each Aij contain q or none of these simplices contain q.

AB - The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ∈ (0,1] such that no matter how we map the vertices of H into ℝd there is a point covered by at least a c(H)-fraction of the simplices induced by the images of its hyperedges. In [18], motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence {Hn}n=1∞ of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which infn≥1 c(Hn) > 0. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of (d + 1)-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant c = c(d). We also show that, for every d, the best value of the constant c = c(d) that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete (d + 1)-uniform hypergraphs with n vertices, as n → ∞. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any d and any ε > 0, there exists K = K(ε,d) ≥ d + 1 satisfying the following condition. For any k ≥ K, for any point q ∈ ℝd and for any finite Borei measure μ on ℝd with respect to which every hyperplane has measure 0, there is a partition ℝd = A1 ∪. .. ∪ Ak into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples Ai1,..., A id+1 have the property that either all simplices with one vertex in each Aij contain q or none of these simplices contain q.

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M3 - Conference contribution

AN - SCOPUS:79955736052

SN - 9780898719932

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1188

EP - 1197

BT - Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011

T2 - 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011

Y2 - 23 January 2011 through 25 January 2011

ER -