Overlap properties of geometric expanders

Jacob Fox, Mikhail Gromov, Vincent Lafforgue, Assaf Naor, János Pach

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationProceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
Pages1188-1197
Number of pages10
StatePublished - May 12 2011
Event22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 - San Francisco, CA, United States
Duration: Jan 23 2011Jan 25 2011

Other

Other22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
CountryUnited States
CitySan Francisco, CA
Period1/23/111/25/11

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Fox, J., Gromov, M., Lafforgue, V., Naor, A., & Pach, J. (2011). Overlap properties of geometric expanders. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 (pp. 1188-1197)