  In 1942 I. J. Schoenberg proved that a function is positive definite in the
unit sphere if and only if this function is a positive linear combination of
the Gegenbauer polynomials. In this paper we extend Schoenberg's theorem for
multivariate Gegenbauer polynomials. This extension derives new positive
semidefinite constraints for the distance distribution which can be applied for
spherical codes.
