Tight 9-designs on two concentric spheres

TL;DR

This paper proves that tight Euclidean 9-designs do not exist on two concentric spheres in dimensions three and higher, implying the nonexistence of certain optimal cubature formulas for symmetric integrals.

Contribution

It establishes the nonexistence of tight Euclidean 9-designs on two concentric spheres in higher dimensions, advancing understanding of spherical designs and cubature formulas.

Findings

No tight Euclidean 9-designs on two concentric spheres for n ≥ 3

Nonexistence of minimum cubature formulas of degree 9 in these settings

Implications for spherical symmetry in numerical integration

Abstract

The main purpose of this paper is to show the nonexistence of tight Euclidean 9-designs on 2 concentric spheres in $\mathbb R^n$ if $n\geq 3.$ This in turn implies the nonexistence of minimum cubature formulas of degree 9 (in the sense of Cools and Schmid) for any spherically symmetric integrals in $\mathbb R^n$ if $n\geq 3.$