Remarks on Hilbert identities, isometric embeddings, and invariant cubature

Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoirs method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small indices with few points are provided, which are used to update Shatalovs table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.

A new approach for the existence problem of minimal cubature formulas based on the Larman-Rogers-Seidel theorem

In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the Moller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral.