Convex Ternary Quartics Are SOS-Convex

TL;DR

This paper proves that convex ternary quartic forms are necessarily sum-of-squares-convex, establishing a convex analogue to Hilbert's classical theorem on nonnegative ternary quartic forms, highlighting the importance of Hessian matrix structure.

Contribution

It establishes that convex ternary quartic forms are sos-convex, providing a new convex analogue to Hilbert's theorem and emphasizing the role of Hessian matrix structure in the proof.

Findings

Convex ternary quartic forms are sos-convex.

Hessian matrix structure is crucial in the proof.

Analogy to Hilbert's theorem on nonnegative forms.

Abstract

We prove that convex ternary quartic forms are sum-of-squares-convex (sos-convex). This result is in a meaningful sense the ``convex analogue'' a celebrated theorem of Hilbert from 1888, where he proves that nonnegative ternary quartic forms are sums of squares. We show by an appropriate construction that exploiting the structure of the Hessian matrix is crucial in any possible proof of our result.