  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.
