AI 中文总结
该研究针对球面$t$-设计关联的闵可夫斯基多面体，证明其具有阶数$t$内的各向同性，给出其豪斯多夫距离等定量球度的误差界，且可扩展到$d$维球面。
AI 中文摘要
设$X_N=\{x_1,\dots,x_N\}\subset \Sph^2$为强度$t\ge2$的球面$t$-设计，$\mu_X$为其经验测度。闵可夫斯基定理将$X_N$关联到$\mathbb{R}^3$中的凸多面体$P_X$，该多面体在平移下唯一，其面法向量为设计节点且所有面面积均为$4\pi/N$，等价于$S_{P_X}=4\pi\mu_X$。我们证明该实现将多项式精确性转化为精确凸几何：$P_X$的归一化表面张量与单位球的归一化表面张量在阶数$t$内一致，混合体积对支撑函数为次数不超过$t$的球面多项式的凸体是精确的。接下来推导定量形状信息：球面杰克逊逼近给出$1$-瓦瑟斯坦偏差$W_1(\mu_X,\sigma)=O(t^{-1})$，二阶精确性给出一致非退化条件。结合闵可夫斯基问题的定量逆稳定性，经施泰纳归一化后，对任意球面$t$-设计，均有$d_H(P_X,B)=O(t^{-1/2})$、$\alpha(P_X,B)=O(t^{-3/4})$，且无需对基数、分离度、覆盖半径或谱条件施加假设。投影体保留完整的$O(t^{-1})$尺度，将线性表面积观测与凸体的非线性重建分离。在临界 regime $N=(t+1)^2$下，采样格拉姆矩阵的一致谱下界进一步迫使面法向量在波长尺度$t^{-1}$处分离。该构造可扩展到$\Sph^d$，具有通用豪斯多夫速率$O(t^{-1/d})$。
英文摘要
Let $X_N=\{x_1,\dots,x_N\}\subset \Sph^2$ be a spherical $t$-design of strength $t\ge2$, and let $μ_X$ be its empirical measure. Minkowski's theorem associates with $X_N$ a convex polytope $P_X\subset\R^3$, unique up to translation, whose facet normals are the design nodes and whose facets all have area $4π/N$; equivalently, $S_{P_X}=4πμ_X$. We show that this realization transfers polynomial exactness into exact convex geometry: the normalized surface tensors of $P_X$ agree with those of the unit ball through order $t$, and mixed volumes are exact against convex bodies whose support functions are spherical polynomials of degree at most $t$. We next derive quantitative shape information. Spherical Jackson approximation yields a $1$-Wasserstein discrepancy $W_1(μ_X,σ)=O(t^{-1})$, while degree-two exactness gives a uniform nondegeneracy condition. Combined with quantitative inverse stability for Minkowski's problem, this implies, after Steiner normalization, \[ d_H(P_X,B)=O(t^{-1/2}),\qquad α(P_X,B)=O(t^{-3/4}), \] for every spherical $t$-design, without assumptions on cardinality, separation, covering radius, or spectral conditioning. Projection bodies retain the full $O(t^{-1})$ scale, separating linear surface-area observables from nonlinear reconstruction of the body. In the critical regime $N=(t+1)^2$, a uniform spectral lower bound for the sampling Gram matrix further forces the facet normals to be separated at the wavelength scale $t^{-1}$. The construction extends to $\Sph^d$, with the universal Hausdorff rate $O(t^{-1/d})$.
