Minkowski Polytopes of Spherical Designs: High-Order Isotropy and Quantitative Sphericity
Let $X_N=\{x_1,\dots,x_N\}\subset \Sph^2$ be a spherical $t$-design of strength $t\ge2$, and let $\mu_X$ 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 $4\pi/N$; equivalently, $S_{P_X}=4\pi\mu_X$. 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 $W_1(\mu_X,\sigma)=O(t^{-1})$, while degree-two exactness gives a uniform nondegeneracy condition. Combined with quantitative inverse stability for Minkowski's problem, this implies, after Steiner normalization, \[ d_H(P_X,B)=O(t^{-1/2}),\qquad \alpha(P_X,B)=O(t^{-3/4}), \] 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 $O(t^{-1/d})$.