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Gauss--Chebyshev Quadrature, $\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry
Congpei An
TL;DR
The paper sets out to place classical circle quadrature and Minkowski reconstruction in a common error framework. It relates quadrature defects to geometric errors and derives inverse-stability and convergence estimates, including the regular grid’s exact N^-2 Hausdorff rate.
Problem
The basic quadrature and reconstruction statements are classical; the paper seeks a common error framework and explicit geometric estimates.
Source: §1 · p. 1Method
A common family of circle quadrature defects is used to express integration, mixed-area, and support-function errors, alongside estimates for the inverse Minkowski map.
Source: §1 · p. 2Results
The regular grid has an exact N^-2 Hausdorff rate, and the inverse-stability exponent is sharp even for smooth positive densities near the uniform measure.
Source: §1 · p. 2, §6.3 · p. 10Takeaways & Limitations
Sufficient conditions on balanced perturbations preserve the regular grid’s quadratic rate and its leading constant.
Source: §1 · p. 2The polygon formula and linear inverse kernel are specific to the plane and imply no higher-dimensional N^-2 reconstruction rate.
Source: §6.3 · p. 11
Abstract
from arXiv · showhide
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 $2\pi$ has the exact Hausdorff distance from the unit disk $B^2$ \[ d_H(P_N,B^2)=1-\frac{\pi}{N}\cot\frac{\pi}{N} =\frac{\pi^2}{3N^2}+O(N^{-4}). \] With Steiner centering, we obtain $d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu)$ 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 $W_1(\nu_N,\sigma)=\pi/(2N)$ 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.
1 Introduction
The paper places circle quadrature, Gauss–Chebyshev quadrature, and planar Minkowski reconstruction in a common error framework. It derives geometric estimates, including inverse stability and the regular grid’s quadratic Hausdorff convergence rate.
1 Introduction
- The regular N-point grid is an S^1-design of strength N −1 and supports reconstruction of a regular planar polygon from its directions.A half-step rotation of the 2M-point grid followed by cosine projection gives the M-point Gauss–Chebyshev rule.Source: §1 · p. 1
- One family of circle quadrature defects expresses Gauss–Chebyshev integration error, mixed-area error, and support-function error.The paper also gives a Lipschitz estimate for the inverse Minkowski map, establishes sharpness of its stability exponent, and derives the regular grid’s exact N^-2 Hausdorff rate.Source: §1 · p. 2
2 Equal spacing and circle designs
The equally spaced N-point grid integrates trigonometric polynomials of degree at most N −1 exactly, making it a sharp S^1-design of strength N −1.
2 Equal spacing and circle designs
- The design condition tests equal-weight averages against restrictions of real polynomials in two variables of total degree at most the strength.On S1 these restrictions are real trigonometric polynomials; the paper uses their complexification for Fourier calculations.Source: §2 · p. 2
- The equally spaced N-point grid is an S^1-design of strength N −1, and this strength is sharp.Its discrete average agrees with the circle integral for frequencies below N; cos(Nθ) demonstrates failure at degree N.Source: §2 · p. 2
3 From the circle rule to Gauss–Chebyshev quadrature
A half-step-rotated equal-weight circle rule projects under cosine to the Gauss–Chebyshev rule. The same quadrature-defect family connects integration error to geometric errors.
3 From the circle rule to Gauss–Chebyshev quadrature
- The M-point Gauss–Chebyshev rule is the even cosine projection of a half-step-rotated 2M-point circle grid and is exact through degree 2M −1.The projected nodes are the M distinct algebraic nodes obtained by pairing the 2M circle nodes.Source: §3 · p. 3
3.1 One family of error functionals
- The half-step rotation is essential for obtaining the Gauss–Chebyshev nodes through cosine projection.The accompanying illustration shows the rotated eight-point grid, its Minkowski polygon with prescribed outward normals, and four projected nodes against the unit circle.Source: §3.1 · p. 4, Figure 1 · p. 4
- One family of quadrature defects links the projected integration error to mixed-area error and support-function error.Testing the defect against support functions gives mixed-area error; testing against translates of the inverse kernel gives support-function error.Source: §3.1 · p. 4
4 The planar Minkowski inverse problem
The planar Minkowski theorem reconstructs a convex body, up to translation, from a balanced surface-area measure. For the regular grid, the reconstruction is a regular polygon, and the quadrature defect also has an exact mixed-area interpretation.
4 The planar Minkowski inverse problem
- The planar Minkowski theorem reconstructs a convex body uniquely up to translation from a positive measure satisfying the balance condition and not concentrated on an antipodal pair.For the regular-grid measure, reconstruction requires N ≥3; N=1 fails balance and N=2 is supported on an antipodal pair.Source: §4 · p. 4
- For N ≥3, the equally spaced directions with mass 2π/N reconstruct, up to translation, a regular N-gon with side length 2π/N.The measure is balanced, and the polygon is centered at the origin for the subsequent comparison.Source: §4 · pp. 4–5
4.1 Quadrature error as mixed-area error
- The quadrature defect for the regular polygon has an exact mixed-area interpretation relative to the unit disk.If hK is a trigonometric polynomial of degree at most N −1, then V(PN,K)=V(B2,K).Source: §4.1 · p. 5
5 Exact Hausdorff convergence to the disk
Theorem 5.1 gives the exact Hausdorff error for the origin-centered regular polygon, and the Fourier multiplier identity explains its quadratic rate through low-frequency cancellation, a sparse discrepancy spectrum, and decay of the inverse multiplier.
5 Exact Hausdorff convergence to the disk
- The Hausdorff comparison uses the origin-centered polygon and unit disk, fixing the translation ambiguity in Minkowski reconstruction.For the centered regular polygon, the support function is smallest normal to a side and largest toward a vertex.Source: §5 · p. 6
- For every N ≥3, Theorem 5.1 gives the exact Hausdorff error for the origin-centered polygon, with the stated asymptotics.The comparison fixes the translation ambiguity by using the origin-centered polygon and the unit disk.Source: §5 · p. 6
5.1 Why the rate is quadratic: a Fourier multiplier identity
- The support-function difference is bounded by summing its Fourier series, and the bound is attained at θ = 0.At θ = 0, the support function equals the polygon’s minimum value r_N.Source: §5.1 · p. 7
- Low-frequency cancellation, discrepancy concentrated on multiples of N, and inverse-multiplier decay like k^-2 explain the quadratic rate.The uniform series estimate makes this mechanism quantitative.Source: §5.1 · p. 7
6 Quantitative stability, discrepancy, and perturbations
This section develops planar inverse stability, quantifies the regular grid’s discrepancy and convergence, and examines sharpness and balanced perturbations. It also states the scope boundary for extending the quadratic rate beyond the plane.
6.1 An explicit inverse kernel
- An explicit Green kernel gives a Lipschitz estimate for the Steiner-centered inverse, with a constant independent of atom counts, atom sizes, and measure concentration.Termwise Fourier bounds are not summable, so estimating the inverse kernel’s Lipschitz norm avoids that loss.Source: §6.1 · pp. 8–9
- One quadrature-error family represents integration, mixed-area, and support-function errors, including an exact support-function error identity for the regular grid.The kernel representation is used to obtain the grid’s support-function identity.Source: §6.1 · p. 9
6.2 Exact discrepancy of the regular grid
- For every N ≥ 1, the regular grid has exact geodesic W1 discrepancy π/(2N); its sharper N^-2 rate follows from sparse grid spectrum and the uniform series bound (4).Source: §6.2 · pp. 9–10
6.3 Sharpness and balanced perturbations
- No uniform Hausdorff bound with W1 exponent β > 1 holds near the uniform measure, even for smooth positive densities converging uniformly to 1.Along the displayed family, dH = (π/3)W1 as a approaches zero.Source: §6.3 · p. 10
- For balanced perturbations, εN = O(N^-2) suffices for dH(KρN, B2) = O(N^-2); necessity of the corollary’s conditions is not asserted.If εN = o(N^-2), the corollary gives a further conclusion, though its formula is not included in the supplied passage.Source: §6.3 · pp. 10–11
- The polygon formula and linear inverse kernel are specific to the plane and imply no higher-dimensional N^-2 reconstruction rate.In ambient dimension at least three, the smooth reconstruction equation is nonlinear.Source: §6.3 · p. 11