Fourth order moments of nonnegative measures on S² and applications

Abstract: "We characterize the set of fourth order moments of a nonnegative measure on S┬▓. This question is motivated by problems of homogenization in linearized elasticity. As a consequence the minimum number of layering directions for generating general anisotropic, transversely isotropic or isotropic multilayered media is found. A key ingredient is a result about the decomposition of some nonnegative polynomials into sums of squares of polynomials; this particular result is due to Hilbert (1888)."