On positive cubature rules on the simplex and isometric embeddings

Positive cubature rules of degree 4 and 5 on the $d$-dimensional simplex are constructed and used to construct cubature rules of index 8 or degree 9 on the unit sphere. The latter ones lead to explicit isometric embedding among the classical Banach spaces. Among other things, our results include several explicit representations of $(x_1^2+\mathrm{.}\mathrm{.}\mathrm{.}+x_d^2{)}^t$ in terms of linear forms of degree $2t$ with rational coefficients for t=4 and 5.