Convexity properties of the cone of nonnegative polynomials

Abstract

We study metric properties of the cone of homogeneous non-negative 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 non-negative 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 (x_1^2 + ... + x_n^2)^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 non-negative 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.