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

TL;DR

This paper investigates the geometric properties of Hankel form cones, specifically the positive semi-definite and sum-of-squares cones, revealing their convexity, dual structures, and implications for the Hilbert-Hankel problem.

Contribution

It characterizes the convexity and dual cones of Hankel form PSD and SOS cones, providing explicit dual formulations and advancing understanding of their geometric structure.

Findings

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.

The dual cone of $HSOS(m,n)$ can be explicitly written, aiding further research.

Abstract

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.