Isometric embed-dings between classical Banach spaces, cubature formulas, and spherical designs

If an isometric embedding l_p^m →l_qⁿ with finite p, q>1 exists, then p=2 and q is an even integer. Under these conditions such an embedding exists if and only if n≥N(m, q) where {Mathematical expression} To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2, q)=q/2+1 (by regular (q+2)-gon), N(3, 4)=6 (by icosahedron), N(3, 6)≥11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound for N(m, q) and obtain a series of concrete values, e.g. N(3, 8)=16 and N(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ∼ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).