  A convex form of degree larger than one is always nonnegative since it
vanishes together with its gradient at the origin. In 2007, Parrilo asked if
convex forms are always sums of squares. A few years later, Blekherman answered
the question in the negative by showing through volume arguments that for high
enough number of variables, there must be convex forms of degree as low as 4
that are not sums of squares. Remarkably, no examples are known to date. In
this paper, we show that all convex forms in 4 variables and of degree 4 are
sums of squares. We also show that if a conjecture of Blekherman related to the
so-called Cayley-Bacharach relations is true, then the same statement holds for
convex forms in 3 variables and of degree 6. These are the two minimal cases
where one would have any hope of seeing convex forms that are not sums of
squares (due to known obstructions). A main ingredient of the proof is the
derivation of certain "generalized Cauchy-Schwarz inequalities" which could be
of independent interest.
