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SCG '88 Proceedings of the fourth annual symposium on Computational geometry
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ACM Computing Surveys (CSUR)
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ESA'11 Proceedings of the 19th European conference on Algorithms
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The overlap number of a finite (d + 1)-uniform hypergraph H is the largest constant c(H) ε (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 [EQUATION] 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 ε Rd and for any finite Borel 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,...,Aid+1 have the property that either all simplices with one vertex in each Aij contain q or none of these simplices contain q.