On optimal tight 4-designs on 2 concentric spheres

Euclidean design (rotatable design) is one of the most important concepts concerning good structures consisting of finitely many points in Rn. Optimal design is an important class of such designs studied in statistics. In this paper we consider Euclidean tight 4-designs XX in Rn which are supported by a union SS of 2 concentric spheres centered at the origin. We prove that if XX is optimal on SS, then 0∈X0∈X and X∖{0}X∖{0} is similar to a spherical tight 4-design.