  We study the quantitative relationship between the cones of nonnegative
polynomials, cones of sums of squares and cones of sums of powers of linear
forms. We derive bounds on the volumes (raised to the power reciprocal to the
ambient dimension) of compact sections of the three cones. We show that the
bounds are asymptotically exact if the degree is fixed and number of variables
tends to infinity. When the degree is larger than two it follows that there are
significantly more non-negative polynomials than sums of squares and there are
significantly more sums of squares than sums of powers of linear forms.
Moreover, we quantify the exact discrepancy between the cones; from our bounds
it follows that the discrepancy grows as the number of variables increases.
