{
  "id": "1007.3546",
  "version": "v2",
  "published": "2010-07-21T01:08:47.000Z",
  "updated": "2010-07-24T00:18:12.000Z",
  "title": "Lower bounds for designs in symmetric spaces",
  "authors": [
    "Noa Eidelstein",
    "Alex Samorodnitsky"
  ],
  "comment": "Abstract and introduction slightly expanded",
  "categories": [
    "math.CO"
  ],
  "abstract": "A design is a finite set of points in a space on which every \"simple\" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower bounds on designs in spaces with a large group of symmetries. These spaces include globally symmetric Riemannian spaces (of any rank) and commutative association schemes with 1-transitive group of symmetries. Our bounds are, in general, implicit, relying on estimates on the spectral behavior of certain symmetry-invariant linear operators. They reduce to the first linear programming bound for designs in globally symmetric Riemannian spaces of rank 1 or in distance regular graphs. The proofs are different though, coming from viewpoint of abstract harmonic analysis in symmetric spaces. As a dividend we obtain the following geometric fact: a design is large because a union of \"spherical caps\" around its points \"covers\" the whole space.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-07-24T00:18:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bounds",
      "symmetric spaces",
      "globally symmetric riemannian spaces",
      "distance regular graphs",
      "abstract harmonic analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.3546E"
    }
  }
}