Gauss--Chebyshev Quadrature, $\mathbb S^1$-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry

We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular $N$-point grid is an $\mathbb S^1$-design of strength $N-1$; a half-step rotation of the $2M$-point grid projects to the classical $M$-point Gauss--Chebyshev ru