On tight Euclidean $6$-designs: an experimental result

TL;DR

This paper proves the uniqueness of certain tight Euclidean 6-designs supported on two spheres in dimensions 2 through 8, building on Bajnok's earlier constructions in the plane.

Contribution

It establishes the uniqueness of tight Euclidean 6-designs supported on two spheres for dimensions 2 to 8, extending Bajnok's planar results.

Findings

Uniqueness of tight Euclidean 6-designs in dimensions 2-8

Extension of Bajnok's planar constructions to higher dimensions

Characterization of configurations supported on two spheres

Abstract

A finite set $X \seq \RR^n$ with a weight function $w : X \longrightarrow \RR_{>0}$ is called \emph{Euclidean $t$-design} in $\RR^n$ (supported by $p$ concentric spheres) if the following condition holds: \[ \sum_{i=1}^p \frac{w(X_i)}{|S_i|}\int_{S_i} f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x \in X}w(\boldsymbol x) f(\boldsymbol x), \] for any polynomial $f(\boldsymbol x) \in \mbox{Pol}(\RR^n)$ of degree at most $t$. Here $S_i \seq \RR^n$ is a sphere of radius $r_i \geq 0,$ $X_i=X \cap S_i,$ and $\sigma_i(\boldsymbol x)$ is an $O(n)$-invariant measure on $S_i$ such that $|S_i|=r_i^{n-1}|S^{n-1}|$, with $|S_i|$ is the surface area of $S_i$ and $|S^{n-1}|$ is a surface area of the unit sphere in $\RR^n$. Recently, Bajnok (2006) constructed tight Euclidean $t$-designs in the plane ($n=2$) for arbitrary $t$ and $p.$ In this paper we show that for case $t=6$ and $p=2,$ tight Euclidean $6$-designs constructed by Bajnok is the unique configuration in $\RR^n$, for $2 \leq n \leq 8.$