{
  "abstract": "The symmetry measure of a convex body $S\\subset\\mathbb{R}^n$ is given by: $\\mathrm{sym}(S):=\\max\\{α\\ge0:\\text{ there exists }x\\in S\\text{ such that }-α(S-x)\\subseteq S-x\\}$, where such an $x$ is called a Minkowski center. We prove that every convex body $S$ admits a $\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$-rounding of $S$, namely, there exists an origin-centered ellipsoid $E$ and a center $c$ such that $E\\subseteq S-c\\subseteq\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}\\,E$. This result was conjectured in 2005 by Belloni and Freund. As special cases, this recovers an $n$-rounding of $S$ (since $\\mathrm{sym}(S)\\ge\\frac{1}{n}$), and a $\\sqrt{n}$-rounding when $\\mathrm{sym}(S)=1$. In the case when $S$ is a polytope given as the convex hull of points, the desired rounding is produced by a regularized minimum-volume covering ellipsoid problem where the regularization is with respect to the Minkowski center. Similarly, when $S$ is a polytope given as the intersection of halfspaces, such a rounding is produced by a regularized maximum-volume inscribed ellipsoid problem. In both of these cases, the rounding can be computed by first solving a linear optimization problem (to compute $\\mathrm{sym}(S)$ and a Minkowski center), and then solving a convex optimization problem with a logarithmic determinant objective, second-order cone constraints, and one semidefinite cone constraint. We also show that the factor $\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$ is nearly tight in its dependence on dimension and symmetry. When $\\frac{n+1}{1+\\mathrm{sym}(S)}$ is an integer, we show by explicit construction that the factor $\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$ is tight. In the more general case, for every dimension $n$ and every admissible symmetry value, we construct a polytope $S$ for which every rounding factor is at least $\\sqrt{\\frac{2}{3}}\\sqrt{\\frac{n}{\\mathrm{sym}(S)}}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30876",
  "author_total": 2,
  "authors": [
    {
      "name": "Zikai Xiong"
    },
    {
      "name": "Robert M. Freund"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66vqhmhxrkcce.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30876",
  "fulltext_url": null,
  "id": "66vqhmhxrkcce",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30876",
  "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",
      "math.OC"
    ],
    [
      "openalex",
      "2604|Point processes and geometric inequalities"
    ]
  ],
  "openalex": "W7204889372",
  "pdf_url": "https://arxiv.org/pdf/2608.30876",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66vqhmhxrkcce.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2608.30876",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30876"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Symmetry-dependence in Rounding of a Convex Body",
  "title_text": "Symmetry-dependence in Rounding of a Convex Body",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/66vqhmhxrkcce",
  "venue": "arXiv",
  "versions": []
}