A new approach for the existence problem of minimal cubature formulas based on the Larman-Rogers-Seidel theorem
This resolves existence questions for minimal cubature formulas in high dimensions, providing a theoretical limitation for numerical integration.
The paper proves that for sufficiently large dimensional minimal cubature formulas of degree 4k+1, points on a concentric sphere have rational inner products, and uses this to show nonexistence of d-dimensional minimal formulas of degrees 13 and 21 for certain integrals when d > 2.
In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the Möller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral.