Overlap properties of geometric expanders

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

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

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 ℝ d 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 {H n} n=1 of arbitrarily large (d + 1)-uniform hypergraphs with bounded degree, for which inf n≥1 c(H n) > 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 ∈ ℝ d and for any finite Borei measure μ on ℝ d with respect to which every hyperplane has measure 0, there is a partition ℝ d = A 1 ∪. .. ∪ A k into k measurable parts of equal measure such that all but at most an ε-fraction of the (d + 1)-tuples A i1,..., A id+1 have the property that either all simplices with one vertex in each A ij 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 - 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

Fingerprint

Expander
Uniform Hypergraph
Overlap
Simplicial Complex
Hyperplane
Partitioning
High-dimensional
Partition
Analogue
Graph in graph theory
Vertex of a graph

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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)

Overlap properties of geometric expanders. / Fox, Jacob; Gromov, Mikhael; Lafforgue, Vincent; Naor, Assaf; Pach, János.

Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011. 2011. p. 1188-1197.

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

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, 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, CA, United States, 1/23/11.
Fox J, Gromov M, Lafforgue V, Naor A, Pach J. Overlap properties of geometric expanders. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011. 2011. p. 1188-1197
Fox, Jacob ; Gromov, Mikhael ; Lafforgue, Vincent ; Naor, Assaf ; Pach, János. / Overlap properties of geometric expanders. Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011. 2011. pp. 1188-1197
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