{
  "id": "2205.10099",
  "version": "v1",
  "published": "2022-05-20T11:30:55.000Z",
  "updated": "2022-05-20T11:30:55.000Z",
  "title": "d-representability as an embedding problem",
  "authors": [
    "Moshe White"
  ],
  "comment": "20 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "An abstract simplicial complex is said to be $d$-representable if it records the intersection patterns of a collection of convex sets in $\\mathbb{R}^d$. In this paper, we show that $d$-representability of a simplicial complex is equivalent to the existence of a map with certain properties, from a closely related simplicial complex into $\\mathbb{R}^d$. This equivalence suggests a framework for proving (and disproving) $d$-representability of simplicial complexes using topological methods such as applications of the Borsuk-Ulam theorem, which we begin to explore.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-20T11:30:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "embedding problem",
      "d-representability",
      "abstract simplicial complex",
      "intersection patterns",
      "convex sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}