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The positive semi-definite cone and sum-of-squares cone of Hankel form

In this paper, the geometry properties of Hankel form are studied, including their positive semi-definite (PSD) cone and sum-of-squares (SOS) cone. We denote them by $HPSD(m,n)$ and $HSOS(m,n)$, respectively. We show that both $HPSD(m,n)$ and $HSOS(m,n)$ are closed convex cones. The dual cone of $HPSD(m,n)$ is the convex hull of all $m$-times convolutions of real vectors. Besides, we derive the dual cone of SOS tensors. By reformulation, it follows that the dual cone of $HSOS(m,n)$ can also be written explicitly. These results may lead further research on the Hilbert-Hankel problem.