On Tight Euclidean 6-Designs: An Experimental Result - Djoko Suprianto (Journal of Mathematical and Fundamental Sciences, 2011)
A finite set n X âŠ† â„n with a weight function w: X â†’ â„ > 0 is called Euclidean t-design in â„ > 0 (supported by p concentric spheres) if the following condition holds: ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© ï€¨ ï€© 1 i p i  i i S x X w X f d w f S ï³ ï€½ ïƒŽ ïƒ¥ ïƒ² x x ï€½ïƒ¥ x x for any polynomial f(x) âˆˆ Polâ„ > 0 of degree at most t. Here Si â„n is a sphere of radius ri â‰¥ 0, Xi=X âˆ© S, and Ïƒi(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 â„n. 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 â„œn, for 2 â‰¤ n â‰¤ 8.