{
  "abstract": "Let $g_n$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $\\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $α_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \\[g_n\\ge (1.160235457\\ldots-o(1))^n,\\] improving the previous best bound $(2/\\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $α_0$ is not attained by any finitely supported distribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "name": "Andrii Arman"
    },
    {
      "name": "Andriy Bondarenko"
    },
    {
      "name": "Andriy Prymak"
    },
    {
      "name": "Danylo Radchenko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5sd6czst51nrf.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "5sd6czst51nrf",
  "issns": [],
  "landing_url": "https://hal.science/hal-05851861v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-mg"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05851861/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5sd6czst51nrf.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:13Z",
      "native_id": "hal-05851861",
      "source": "hal",
      "url": "https://hal.science/hal-05851861v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-mg",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Grünbaum's problem for symmetric configurations",
  "title_text": "On Grünbaum's problem for symmetric configurations",
  "type": "preprint",
  "updated_at": "2026-11-06T12:20:44Z",
  "url": "https://research.draconitic.com/paper/5sd6czst51nrf",
  "venue": "HAL",
  "versions": []
}