The Grasshopper Problem on the Sphere

The spherical grasshopper problem is a geometric optimization problem that arises in the context of Bell inequalities and can be interpreted as identifying the best local hidden variable approximation to quantum singlet correlations for measurements along random axes separated by a fixed angle. In a parallel publication [arXiv:2504.20953], we presented numerical solutions for this problem and explained their significance for singlet simulation and testing. In this companion paper, we describe in detail the geometric and computational framework underlying these results. We examine the role of spherical discretization and compare three natural variants of the problem: antipodal complementary lawns, antipodal independent lawns, and non-antipodal complementary lawns. We analyze the geometric structure of the corresponding optimal lawn configurations and interpret it in terms of a spherical harmonics expansion. We also discuss connections to other physical models and to classical problems in geometric probability.