Overlap properties of geometric expanders

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 = A₁ ∪. .. ∪ 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.