Abstract

We study metric properties of the cone of homogeneousnonnegative multivariate polynomials and the cone of sums ofpowers of linear forms, and the relationship between the twocones. We compute the maximum volume ellipsoid of the natural baseof the cone of nonnegative polynomials and the minimum volumeellipsoid of the natural base of the cone of powers of linearforms and compute the coefficients of symmetry of the bases. Themultiplication by m induces anisometric embedding of the space of polynomials of degree 2kinto the space of polynomials of degree 2, which allows usto compare the cone of nonnegative polynomials of degree 2k andthe cone of sums of 2-powers of linear forms. We estimatethe volume ratio of the bases of the two cones and the rate atwhich it approaches 1 as m grows.