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

A Gaussian tt-design is defined as a finite set XX in the Euclidean space RnRn satisfying the condition: 1V(Rn)∫Rnf(x)e−α2‖x‖2dx=∑x∈Xω(x)f(x) for any polynomial f(x)f(x) in nn variables of degree at most tt, where αα is a constant real number and ωω is a positive weight function on XX. It is well known that if XX is a Gaussian 2e2e-design in RnRn, then |X|≥n+ee. We call XX a tight Gaussian 2e2e-design in RnRn if |X|=n+ee. In this paper, we prove that there exists no tight Gaussian 6-design supported by two concentric spheres in RnRn for n≥2n≥2.