AI 中文总结
本文研究球面 $t$-设计的对偶几何，推导自极球面设计的关联水平下界，证明近等条件下的刚性结果，确定正则单纯形和24胞腔为正则自极球面设计。
AI 中文摘要
设 $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ 是一个球面 $t$-设计，其中 $t\ge2$，$P_X\subset\R^d$ 是其闵可夫斯基多面体。若 $h_i=h_{P_X}(x_i)$，则 $P_X^\circ=\conv\{x_i/h_i:1\le i\le N\}$，即该设计是极顶点集的径向投影，由此可得到精确的依赖于次数的矩恒等式，并从 $P_X$ 的豪斯多夫球度出发，对无权重极矩进行定量控制。本文主要结果涉及自极性：若 $P_X^\circ=cUP_X$（其中 $U\in O(d)$），则可得到结构化松弛分解 $A=c\\,hh^T-X^TU^TX$，其秩为 $d+1$，该分解的零模式记录了面-顶点关联关系。对于节点可递设计，公共关联水平等于内切半径与外接半径之比 $r/R$。将该结果与 Fazekas-Levenshtein 覆盖界所基于的一维矩问题相结合，可证明 $\frac rR\ge\eta_{t,d}$，当等号成立时，会迫使扭曲内积行实现对应的高斯或高斯-拉道求积规则；特别地，量 $N\lambda_k$ 成为整数关联重数。本文进一步证明了一个定量近等定理：若 $\delta=\frac rR-\eta_{t,d}$ 很小，则每个行在 $W_1$ 中与极值求积测度的距离为 $O_{d,t}(\sqrt\delta)$，从而得到近关联刚性与算术稳定性间隙。作为应用，本文得到了三维刚性结果，并确定正则单纯形和 $24$ 胞腔是正则自极示例。
英文摘要
Let $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ be a spherical $t$-design, $t\ge2$, and let $P_X\subset\R^d$ be its Minkowski polytope. If $h_i=h_{P_X}(x_i)$, then \[ P_X^\circ=\conv\{x_i/h_i:1\le i\le N\}, \] so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of $P_X$. Our main results concern self-polarity. If \[ P_X^\circ=cUP_X,\qquad U\in O(d), \] we obtain the structured slack factorization \[ A=c\,hh^T-X^TU^TX,\qquad \rank A=d+1, \] whose zero pattern records the facet--vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio $r/R$. Combining this with the one-dimensional moment problem underlying the Fazekas--Levenshtein covering bound, we prove \[ \frac rR\geη_{t,d}, \] with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss--Radau quadrature rule; in particular, the quantities $Nλ_k$ become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if \[ δ=\frac rR-η_{t,d} \] is small, then each row is $O_{d,t}(\sqrtδ)$-close in $W_1$ to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.
