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PreprintCC BY 4.0arXiv

Gauss--Chebyshev Quadrature, 𝕊1-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry

  • Congpei An
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Abstract

We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular N-point grid is an 𝕊1-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≥3, the centered Minkowski polygon PN with perimeter 2π has the exact Hausdorff distance from the unit disk B2 dH(PN,B2)=1−π/Ncotπ/N =π2/3N2+O(N−4). With Steiner centering, we obtain dH(Kμ,Kν)≤πW1(μ,ν) for balanced, nondegenerate probability measures, where W1 uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity W1(νN,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−k2)−1. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.

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Cite

BibTeXRIS
BibTeX
@misc{an2026gausschebyshev,
  title = {{Gauss--Chebyshev Quadrature, $\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry}},
  author = {Congpei An},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11814},
  eprint = {2610.11814},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11814},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Gauss--Chebyshev Quadrature, 𝕊^1-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry
AU  - Congpei An
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11814
UR  - https://arxiv.org/abs/2610.11814
AB  - We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular N-point grid is an 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≥3, 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−π/Ncotπ/N =π^2/3N^2+O(N^−4). With Steiner centering, we obtain d_H(K_μ,K_ν)≤πW_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.
ER  -