On the non-existence of tight Gaussian 6-designs on two concentric spheres

A Gaussian t -design is defined as a finite set X in the Euclidean space R n satisfying the condition: 1 V ( R n ) ∫ R n f ( x ) e − α 2 ‖ x ‖ 2 d x = ∑ x ∈ X ω ( x ) f ( x ) for any polynomial f ( x ) in n variables of degree at most t , where α is a constant real number and ω is a positive weight function on X . It is well known that if X is a Gaussian 2 e -design in R n , then | X | ≥ n + e e . We call X a tight Gaussian 2 e -design in R n if | X | = n + e e . In this paper, we prove that there exists no tight Gaussian 6-design supported by two concentric spheres in R n for n ≥ 2 .