On Tight Euclidean 6-Designs: An Experimental Result

Abstract

A finite set n X ⊆ ℝn with a weight function w: X → ℝ > 0 is called Euclidean t-design in ℝ > 0 (supported by p concentric spheres) if the following condition holds:           1 i p i i i S x X w X f d w f S      x x  x x for any polynomial f(x) ∈ Polℝ > 0 of degree at most t. Here Si ℝn is a sphere of radius ri ≥ 0, Xi=X ∩ S, and σi(x) is an O(n) -invariant measure on Si such that |Si|=rin-1|Sn-1>|, with |Si| is the surface area of Si and |Sn-1|is a surface area of the unit sphere in ℝn. Recently, Bajnok [1] 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 ℜn, for 2 ≤ n ≤ 8.