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

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-depen