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

TL;DR

This paper investigates the existence of minimal cubature formulas of certain degrees for spherically symmetric integrals, establishing rationality conditions on point inner products and proving non-existence results in specific dimensions.

Contribution

It introduces a novel approach based on the Larman-Rogers-Seidel theorem to analyze minimal cubature formulas and proves their non-existence for degrees 13 and 21 in dimensions greater than 2.

Findings

Minimal formulas require rational inner products on some spheres.

No minimal formulas of degrees 13 and 21 exist for certain integrals in dimensions > 2.

Abstract

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 M\"oller 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.