Convexity Properties of The Cone of Nonnegative Polynomials

We study metric properties of the cone of homogeneous nonnegative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of nonnegative polynomials and the minimum volume ellipsoid of the natural base of the cone of powers of linear forms and compute the coefficients of symmetry of the bases. The multiplication by (x₁²+...+x_n²)^m induces an isometric embedding of the space of polynomials of degree 2k into the space of polynomials of degree 2(k+m), which allows us to compare the cone of nonnegative polynomials of degree 2k and the cone of sums of 2(k+m)-powers of linear forms. We estimate the volume ratio of the bases of the two cones and the rate at which it approaches 1 as m grows.