Cubature Formulae, Polytopes, and Spherical Designs

The construction of a cubature formula of strength t for the unit sphere Ω d in ℝ d amounts to finding finite sets X 1,..., X N ⊂ Ω d and coefficients a 1,..., a N ∈ ℝ such that 1.1 $$ |{\Omega _d}{|^{ - 1}}\int_{{\Omega _d}}^{} {f(\xi )d\omega } (\xi ) = \sum\limits_{i = 1}^N {{a_i}|{X_i}{|^{ - 1}}} \sum\limits_{x \in {X_\iota }} {f(x),} $$ for all functions f represented on Ω d by polynomials of degree ⩽ t; cf. [16], [15], [11]. Sobolev [14,15] introduced group theory into the construction of cubature formulae by considering orbits X ι under a finite subgroup G of the orthogonal group O(d). Thus spherical polytopes and root systems (cf. Coxeter [3]) enter the discussion. There are further relations to Coxeter’s work, since the obstruction to higher strength for a cubature formula is caused essentially by the existence of certain invariants. For finite groups generated by reflections, the theory of exponents and invariants goes back to Coxeter [4].