AI 中文总结
本研究探讨等权圆求积、高斯-切比雪夫求积与平面闵可夫斯基重构的关联，利用正则网格的设计性质推导求积误差、混合面积误差等的精确恒等式，得到相关误差界及扰动估计，为设计诱导凸几何提供一维模型。
AI 中文摘要
我们研究等权圆求积、高斯-切比雪夫求积与平面闵可夫斯基重构之间的关联。正则$N$点网格是强度为$\mathbb S^1$的$N-1$设计；$2M$点网格的半步旋转可投影为经典的$M$点高斯-切比雪夫规则。一类公共求积误差泛函给出了积分误差、混合面积误差及支撑函数误差的精确恒等式。对于$N\ge3$，周长为$2\pi$的中心闵可夫斯基多边形$P_N$与单位圆盘$B^2$的精确豪斯多夫距离为：\\[ d_H(P_N,B^2)=1-\frac{\pi}{N}\cot\frac{\pi}{N} =\frac{\pi^2}{3N^2}+O(N^{-4}). \\] 结合施泰纳中心化，我们得到了平衡非退化概率测度的$d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu)$，其中$W_1$使用测地距离。即使对于均匀测度附近的光滑正密度，其指数也是最优的。恒等式$W_1(\nu_N,\sigma)=\pi/(2N)$给出了正则网格的一般$O(N^{-1})$界。更精确的$O(N^{-2})$速率源于稀疏偏差谱和逆乘子$(1-k^2)^{-1}$。最后，扰动估计给出了保持二次速率及其主项常数的充分条件。
英文摘要
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.
