{
  "id": "0710.2162",
  "version": "v1",
  "published": "2007-10-11T04:28:13.000Z",
  "updated": "2007-10-11T04:28:13.000Z",
  "title": "Construction and Analysis of Projected Deformed Products",
  "authors": [
    "Raman Sanyal",
    "Günter M. Ziegler"
  ],
  "comment": "20 pages, 5 figures",
  "journal": "Discrete Comput. Geom. 43 (2010), no. 2, 412-435",
  "doi": "10.1007/s00454-009-9146-6",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representations of deformed products of(even) polygons (DPPs), which have a projection to d-space that retains the complete (\\lfloor \\tfrac{d}{2} \\rfloor - 1)-skeleton. In both cases the combinatorial structure of the images under projection is completely determined by the neighborly polytope Q: Our analysis provides explicit combinatorial descriptions. This yields a multitude of combinatorially different neighborly cubical polytopes and DPPs. As a special case, we obtain simplified descriptions of the neighborly cubical polytopes of Joswig & Ziegler (2000) as well as of the ``projected deformed products of polygons'' that were announced by Ziegler (2004), a family of 4-polytopes whose ``fatness'' gets arbitrarily close to 9.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-11T04:28:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "52B35",
      "52B05"
    ],
    "keywords": [
      "projected deformed products",
      "construction",
      "lower-triangular block matrix representations",
      "projection",
      "neighborly cubical polytopes"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.2162S"
    }
  }
}