Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure

Abstract

Let $X=\{x_1,\ldots,x_N\}\subset\Sph^{d-1}$ be a spherical $t$-design, $t\ge 2$, and let $P_X\subset {\mathbb{R}}^d$ 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 $N{\lambda }_k$ 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 $O_{d,t}(\sqrt{\delta })$-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.