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The cone $\mathcal{P}_{n+1,2d}$ ($n,d\in\mathbb{N}$) of all positivesemidefinite (PSD) real forms in $n+1$ variables of degree $2d$ contains thesubcone $\Sigma_{n+1,2d}$ of those that are representable as finite sums ofsquares (SOS) of real forms of half degree $d$. In 1888, Hilbert proved thatthese cones coincide exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or$2d=2$ or $(n+1,2d)=(3,4)$. To establish the strict inclusion$\Sigma_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ in any non-Hilbert case, onecan show that verifying the assertion in the basic non-Hilbert cases $(4,4)$and $(3,6)$ suffices.  In this paper, we construct a filtration of intermediate cones between$\Sigma_{n+1,2d}$ and $\mathcal{P}_{n+1,2d}$. This filtration is induced viathe Gram matrix approach (by Choi, Lam and Reznick) on a filtration ofirreducible projective varieties $V_{k-n}\subsetneq \ldots \subsetneq V_n\subsetneq \ldots \subsetneq V_0$ containing the Veronese variety. Here, $k$ isthe dimension of the vector space of real forms in $n+1$ variables of degree$d$. By showing that $V_0,\ldots,V_n$ are varieties of minimal degree, wedemonstrate that the corresponding intermediate cones coincide with$\Sigma_{n+1,2d}$. Likewise, for the special case when $n=2$, $V_{n+1}$ is alsoa variety of minimal degree and the corresponding intermediate cone alsocoincides with $\Sigma_{n+1,2d}$. We moreover prove that, in the non-Hilbertcases of $(n+1)$-ary quartics for $n\geq 3$ and $(n+1)$-ary sextics for $n\geq2$, all the remaining cone inclusions are strict.

Intermediate Cones between the Cones of Positive Semidefinite Forms and Sums of Squares