{
  "id": "1005.1392",
  "version": "v1",
  "published": "2010-05-09T12:41:06.000Z",
  "updated": "2010-05-09T12:41:06.000Z",
  "title": "Overlap properties of geometric expanders",
  "authors": [
    "Jacob Fox",
    "Mikhail Gromov",
    "Vincent Lafforgue",
    "Assaf Naor",
    "Janos Pach"
  ],
  "categories": [
    "math.CO",
    "cs.CG"
  ],
  "abstract": "The {\\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\\in (0,1]$ 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~\\cite{Gro2}, 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 $\\{H_n\\}_{n=1}^\\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\\inf_{n\\ge 1} c(H_n)>0$. 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 $n\\rightarrow\\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\\epsilon>0$, there exists $K=K(\\epsilon,d)\\ge d+1$ satisfying the following condition. For any $k\\ge K$, for any point $q \\in \\mathbb{R}^d$ and for any finite Borel measure $\\mu$ on $\\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\\mathbb{R}^d=A_1 \\cup \\ldots \\cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-05-09T12:41:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform hypergraph",
      "overlap properties",
      "geometric expanders",
      "overlap number",
      "higher dimensional simplicial complexes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.1392F"
    }
  }
}