On the existence and non-existence of spherical m-stiff configurations

This paper investigates the existence of m-stiff configurations in the unit sphere Sᵈ⁻¹, which are spherical $(2m-1)$-designs that lie on m parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no m-stiff configuration in Sᵈ⁻¹ for sufficiently large d; (2) for each fixed integer $d > 10$, there exists no m-stiff configuration in Sᵈ⁻¹ for sufficiently large m. Furthermore, we provide a complete classification of the dimensions where m-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of m-stiff configurations in Sᵈ⁻¹ for small d (3 ≤ d ≤ 120) with arbitrary m, and also for small m (6 ≤ m ≤ 10) with arbitrary d. Finally, we conjecture that there is no m-stiff configuration in Sᵈ⁻¹ for $(d,m)$ with d≥ 3 and m≥ 6.