{
  "abstract": "We show that every polytope $P\\subseteq[0,1]^n$, and more generally every compact convex set, has Chvátal rank at most $12.22n^2+n\\log_2 n+2n+4$. This improves the $O(n^2\\log n)$ bound of Eisenbrand and Schulz and, together with the $Ω(n^2)$ lower bound of Rothvoß and Sanità, shows that the maximum Chvátal rank of a polytope in $[0,1]^n$ is $Θ(n^2)$. More precisely, if $P$ contains an integer point, then for every $c\\in\\mathbb{Z}^n\\setminus\\{0\\}$ the inequality $cx\\le\\max\\{cy: y\\in P\\cap\\mathbb{Z}^n\\}$ is valid for the $k$-th Chvátal closure of $P$ for some $k\\le 12.22n^2+2n+2\\log_2\\|c\\|_\\infty+4$. Following Eisenbrand and Schulz, we derive this inequality along a chain of coarser and coarser integer vectors, but instead of halving the vector in each step, we round $τc$ for a scale $τ$ chosen freely in each dyadic window $[2^{-t-1},2^{-t}]$. The main new ingredient is a multiscale version of Dirichlet's approximation theorem, proved by an elementary volume argument: for every $c\\in\\mathbb{R}^n$, the $\\ell_1$-distances of $τc$ to $\\mathbb{Z}^n$, minimized within each dyadic window and summed over all windows, total less than $1.222n^2$, independently of $\\|c\\|_\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32210",
  "author_total": 1,
  "authors": [
    {
      "name": "Alberto Del Pia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f81nzxkj74be5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32210",
  "fulltext_url": null,
  "id": "f81nzxkj74be5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32210",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7214722038",
  "pdf_url": "https://arxiv.org/pdf/2609.32210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f81nzxkj74be5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "A quadratic upper bound on the Chvátal rank of polytopes in the 0/1-cube",
  "title_text": "A quadratic upper bound on the Chvátal rank of polytopes in the 0/1-cube",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/f81nzxkj74be5",
  "venue": "arXiv",
  "versions": []
}