{
  "abstract": "We provide sharp estimates on the number of vertices needed to approximate the Euclidean unit ball $B_d\\subset\\mathbb R^d$ with relative symmetric-difference volume error at most $\\varepsilon\\in(0,1)$. We show that the minimum vertex budget $N_\\varepsilon(d)$ satisfies $$\\log N_\\varepsilon(d)\\asymp d\\log\\left(1+\\frac1{\\varepsilon\\sqrt d}\\right).$$ In particular, $\\exp(O(\\sqrt d/\\varepsilon))$ vertices suffice for every $\\varepsilon\\in(0,1)$ and all sufficiently large $d$. We also prove that achieving a fixed error $\\varepsilon\\in(0,1)$ requires at least $Ω\\bigl((\\tfrac{1+\\varepsilon}{2\\sqrt\\varepsilon})^{d-1}\\bigr)$ facets, an exponential number in $d$. We extend our vertex approximation result to Gaussian symmetric-difference error, and give a positive answer to the question of De, Nadimpalli, and Servedio about subexponential vertex complexity for approximating Euclidean balls. We also disprove a Macbeath-type extremality principle for unrestricted vertex approximation. Whereas the ball uniformly maximizes the inscribed approximation error among all convex bodies (Macbeath, 1951), we show that approximating the hypercube $[-1,1]^d$ to relative error $\\varepsilon$, with $δ:=1-\\varepsilon$, requires $N\\ge\\fracδ{2}\\exp(Ω(δ^2d/\\log^2(2/δ)))$ vertices whenever $d\\ge16\\log(2/δ)$. Thus, when $d$ is large, the cube is harder to approximate than the ball in this sense. Finally, we determine the sharp dimensional order of the classical Laguerre-Delone triangulation number, showing that $\\mathrm{ldel}_{d-1}=Θ(d^{-1/2})$ as $d\\to\\infty$. This improves both the upper bound of Ludwig, Schütt, and Werner and the lower bound of Böröczky by a factor of order $\\sqrt d$. By classical asymptotic approximation theorems (Ludwig, 1999), this improvement applies to every convex body in $\\mathbb R^d$ with $C^2$ boundary and positive Gaussian curvature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34651",
  "author_total": 3,
  "authors": [
    {
      "name": "Yegor Gorodzha"
    },
    {
      "name": "Steven Hoehner"
    },
    {
      "name": "Gil Kur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3zf5rg7rx9qv6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34651",
  "fulltext_url": null,
  "id": "3zf5rg7rx9qv6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34651",
  "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.CG"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7223879170",
  "pdf_url": "https://arxiv.org/pdf/2610.34651",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3zf5rg7rx9qv6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34651",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34651"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions",
  "title_text": "Approximating the Euclidean Ball by Polytopes: Surprising Rates in High Dimensions",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:45Z",
  "url": "https://research.draconitic.com/paper/3zf5rg7rx9qv6",
  "venue": "arXiv",
  "versions": []
}