{
  "abstract": "We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular $N$-point grid is an $\\mathbb S^1$-design of strength $N-1$; a half-step rotation of the $2M$-point grid projects to the classical $M$-point Gauss--Chebyshev rule. A common family of quadrature-error functionals gives exact identities for integration error, mixed-area error, and support-function error. For $N\\ge3$, the centered Minkowski polygon $P_N$ with perimeter $2π$ has the exact Hausdorff distance from the unit disk $B^2$ \\[ d_H(P_N,B^2)=1-\\fracπ{N}\\cot\\fracπ{N} =\\frac{π^2}{3N^2}+O(N^{-4}). \\] With Steiner centering, we obtain $d_H(K_μ,K_ν)\\leπW_1(μ,ν)$ for balanced, nondegenerate probability measures, where $W_1$ uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity $W_1(ν_N,σ)=π/(2N)$ yields a general $O(N^{-1})$ bound for the regular grid. The sharper $O(N^{-2})$ rate follows from the sparse discrepancy spectrum and the inverse multiplier $(1-k^2)^{-1}$. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11814",
  "author_total": 1,
  "authors": [
    {
      "name": "Congpei An"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4mfp07mcwv2m2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11814",
  "fulltext_url": null,
  "id": "4mfp07mcwv2m2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11814",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2612|Mathematical Approximation and Integration"
    ]
  ],
  "openalex": "W7221143751",
  "pdf_url": "https://arxiv.org/pdf/2610.11814",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4mfp07mcwv2m2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11814",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11814"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Gauss--Chebyshev Quadrature, $\\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry",
  "title_text": "Gauss--Chebyshev Quadrature, 𝕊^1-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/4mfp07mcwv2m2",
  "venue": "arXiv",
  "versions": []
}