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

On Sum of Squares Representation of Convex Forms and Generalized Cauchy-Schwarz Inequalities