This paper investigates the existence of m-stiff configurations in the unit sphere S^d-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 S^d-1 for sufficiently large d; (2) for each fixed integer d > 10, there exists no m-stiff configuration in S^d-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 S^d-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 S^d-1 for (d, m) with d 3 and m 6.
Bannaĭ et al. (Thu,) studied this question.