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 R n . Optimal design is an important class of such designs studied in statistics. In this paper we consider Euclidean tight 4-designs X in R n which are supported by a union S of 2 concentric spheres centered at the origin. We prove that if X is optimal on S , then 0 ∈ X and X ∖ { 0 } is similar to a spherical tight 4-design.