  An orbitope is the convex hull of an orbit of a point under the action of a
compact group. We derive bounds on volumes of sections of polar bodies of
orbitopes, extending our previously developed methods. As an application we
realize the cone of convex forms as a section of the cone of nonnegative
bi-homogeneous forms and estimate its volume. A convex form has to be
nonnegative, but it has not been previously shown that there exist convex forms
that are not sums of squares. Combining with the known bounds we show that if
the degree is fixed then the cone of convex forms has asymptotically same size
as the cone of nonnegative forms and it is significantly larger asymptotically
than the cone of sums of squares. This implies existence of convex forms that
are not sums of squares, although there are still no known examples.
