A new approach for the existence problem of minimal cubature formulas based on the Larman-Rogers-Seidel theorem

摘要

In this paper we consider the existence problem of minimal cubature formulas of degree 4k+1 for spherically symmetric integrals. We prove that, for a minimal formula in sufficiently high-dimensional space, there exists some concentric sphere on which the inner product of any two distinct points is rational. By using 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.