Lower bounds for projective designs, cubature formulas and related isometric embeddings

Yudin’s lower bound [V.A. Yudin, Lower bounds for spherical designs, Izv. Math. 61 (3) (1997) 673–683] for the spherical designs is generalized to the cubature formulas on the projective spaces over a field K ∈ { R , C , H } , and thus to isometric embeddings l 2 ; K m → l p ; K n with p ∈ 2 N . For large p and in some other situations this result is substantially better than known before.