{
  "abstract": "We investigate when a parallelotope containing vertices of the unit cube $Q_n=[0,1]^n$ covers $\\Bbb R^n$ by integer translates. We prove the covering property for every parallelepiped containing five vertices of $Q_3$, and, in any dimension, for every parallelotope containing a facet of $Q_n$ and one vertex of the opposite facet. As a consequence, we obtain the sharp bound $1/n$ for the volume of a pyramid contained in $Q_n$ with a parallelotope base. We also give an alternative proof of this bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21572",
  "author_total": 2,
  "authors": [
    {
      "name": "Dragomir Grozev"
    },
    {
      "name": "Nikolai Nikolov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0skbahendf0n4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21572",
  "fulltext_url": null,
  "id": "0skbahendf0n4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21572",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222566533",
  "pdf_url": "https://arxiv.org/pdf/2610.21572",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0skbahendf0n4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21572",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21572"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Lattice coverings by parallelotopes containing cube vertices",
  "title_text": "Lattice coverings by parallelotopes containing cube vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/0skbahendf0n4",
  "venue": "arXiv",
  "versions": []
}