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

2025/04/24 by Bannai, Eiichi, Kurihara, Hirotake, Nozaki, Hiroshi · 1 citation
#05B30 #33C45 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.17184

Abstract

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.

Citations

• Absolute Minima of Potentials of a Certain Class of Spherical Designs
• Universal minima of discrete potentials for sharp spherical codes
• On polarization of spherical codes and designs
• On tight spherical designs
• Tight 9-designs on two concentric spheres
• A New Euclidean Tight 6-Design
• Universally optimal distribution of points on spheres
• Orthogonal Polynomials

Cited by

• Existence and nonexistence of spherical 5-designs of minimal type

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