  We study dimensions of the faces of the cone of nonnegative polynomials and
the cone of sums of squares; we show that there are dimensional differences
between corresponding faces of these cones. These dimensional gaps occur in all
cases where there exist nonnegative polynomials that are not sums of squares.
The gaps occur generically, they are not the product of selecting special faces
of the cones. For ternary forms and quaternary quartics, we completely
characterize when these differences are observed. Moreover, we provide an
explicit description for these differences in the two smallest cases, in which
the cone of nonnegative polynomials and the cone of sums of squares are
different. Our results follow from more general results concerning the
relationship between the second ordinary power and the second symbolic power of
the vanishing ideal of points in projective space.
