{
  "abstract": "Can one construct a graph $G$ on the set of integer points ${\\mathbb Z}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\\ldots,$ every $ρ$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $cρ$-fat. (A convex set is $ρ$-fat if the ratio of its inradius to its circumradius is at least $ρ$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\\mathbb Z}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5/6})$. The same bound can be achieved by a planar graph with vertex set ${\\mathbb Z}^2$, in which every edge joins two vertices at Euclidean distance at most 2.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20702",
  "author_total": 2,
  "authors": [
    {
      "name": "János Pach"
    },
    {
      "name": "Gábor Tardos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ev6v0d2hy5jn5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20702",
  "fulltext_url": null,
  "id": "ev6v0d2hy5jn5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20702",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1704|Computational Geometry and Mesh Generation"
    ]
  ],
  "openalex": "W7213600248",
  "pdf_url": "https://arxiv.org/pdf/2609.20702",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ev6v0d2hy5jn5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20702",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20702"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cutting a convex body into fat parts and approximating Euclidean distance by graph distances",
  "title_text": "Cutting a convex body into fat parts and approximating Euclidean distance by graph distances",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/ev6v0d2hy5jn5",
  "venue": "arXiv",
  "versions": []
}